Lower bounds on the Erdős-Gyárfás problem via color energy graphs
aa r X i v : . [ m a t h . C O ] F e b Lower Bounds on the Erd˝os-Gy´arf´as Problem via ColorEnergy Graphs
J´ozsef Balogh ∗ Sean English † Emily Heath ‡ Robert A. Krueger § Abstract
Given positive integers p and q , a ( p, q )-coloring of the complete graph K n is anedge-coloring in which every p -clique receives at least q colors. Erd˝os and Shelahposed the question of determining f ( n, p, q ), the minimum number of colors neededfor a ( p, q )-coloring of K n . In this paper, we expand on the color energy techniqueintroduced by Pohoata and Sheffer to prove new lower bounds on this function, mak-ing explicit the connection between bounds on extremal numbers and f ( n, p, q ). Usingresults on the extremal numbers of subdivided complete graphs, theta graphs, and sub-divided complete bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer,giving the first nontrivial lower bounds on f ( n, p, q ) for some pairs ( p, q ) and improvingprevious lower bounds for other pairs. Keywords: generalized Ramsey, color energy, local properties
The
Ramsey number r k ( p ) is the minimum number of vertices n for which every edge-coloringof the complete graph K n with k colors must contain a monochromatic copy of the clique K p .In 1975, Erd˝os and Shelah [4, 5] introduced the following natural extension of the Ramseynumber. Given positive integers p and q with 1 ≤ q ≤ (cid:0) p (cid:1) , a ( p, q ) -coloring of K n is an ∗ Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA,and Moscow Institute of Physics and Technology, Russian Federation. E-mail: [email protected] . Re-search supported by NSF RTG Grant DMS-1937241, NSF Grant DMS-1764123, Arnold O. Beckman Re-search Award (UIUC Campus Research Board RB 18132), the Langan Scholar Fund (UIUC), and the SimonsFellowship. † Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.E-mail: [email protected] . ‡ Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.Email: [email protected] . Research partially supported by NSF RTG Grant DMS-1937241. § Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.Email: [email protected] . Research partially supported by NSF RTG Grant DMS-1937241. K n in which every p vertices span a clique with at least q colors. The ( p, q ) -coloring number of K n , denoted by f ( n, p, q ), is the minimum number of colors needed togive a ( p, q )-coloring of K n . Here and throughout, we analyze the asymptotics of f ( n, p, q )in n , considering p , q , and related parameters to be constant. The asymptotic notationsuppresses dependencies on these constants. It is worth noting that determining the valuesof f ( n, p,
2) for all n and p is equivalent to determining the values of the multicolor Ramseyfunction r k ( p ) for all k and p .These generalized Ramsey numbers were first studied systematically by Erd˝os and Gy´arf´as [6].In addition to studying several small cases of p and q , they also identified the values of q asa function of p for which f ( n, p, q ) becomes linear in n , quadratic in n , and asymptoticallyequivalent to (cid:0) n (cid:1) . Namely, they showed that q = (cid:0) p (cid:1) − p +3 is the smallest value of q for which f ( n, p, q ) = Ω( n ) and q = (cid:0) p (cid:1) − ⌊ p ⌋ + 2 is the smallest value of q for which f ( n, p, q ) = Ω( n ).They also applied the Lov´asz Local Lemma [7] to obtain what is still the best known upperbound on f ( n, p, q ) for general p and q , f ( n, p, q ) = O n p − ( p ) − q +1 ! . (1)Additionally, a simple inductive argument was given in [6] to prove a lower bound in thediagonal case, i.e. when p = q , f ( n, p, p ) ≥ n p − − . Since then, significant progress has been made towards understanding the behavior of this( p, q )-coloring function. S´ark¨ozy and Selkow [16, 17] explored the behavior of f ( n, p, q ) forvalues of q between the thresholds given in [6]. In particular, they proved that there are atmost log p values of q for which f ( n, p, q ) is linear, and showed that f ( n, p, q ) = (cid:0) n (cid:1) − o ( n )for all q > (cid:0) p (cid:1) − ⌊ p ⌋ + 2 + ⌈ log p ⌉ . In addition, Conlon, Fox, Lee, and Sudakov [2] provedthat f ( n, p, p −
1) = n o (1) for all p , settling the question posed in [6] of whether q = p is thesmallest value for which f ( n, p, q ) is polynomial in n .However, there are many values of ( p, q ) for which very little is known about f ( n, p, q ).Recently, Pohoata and Sheffer [15] introduced the color energy of a graph, which they usedto obtain lower bounds on the generalized Ramsey numbers for a new family of values of( p, q ).Given a graph G = ( V, E ), a set of colors C , and an edge-coloring χ : E → C , the colorenergy of G is E ( G ) = |{ ( v , v , v , v ) ∈ V : χ ( v , v ) = χ ( v , v ) }| . By bounding the color energy of K n under a ( k, (cid:0) k (cid:1) − m · (cid:4) km +1 (cid:5) + m + 1)-coloring, Pohoataand Sheffer [15] proved the following new family of bounds: Theorem 1.1 ([15]) . For any integers k > m ≥ , f (cid:18) n, k, (cid:18) k (cid:19) − m · (cid:22) km + 1 (cid:23) + m + 1 (cid:19) = Ω (cid:16) n m (cid:17) .
2n [9], Fish, Pohoata, and Sheffer further developed the color energy approach, defininghigher color energies and color energy graphs as tools to prove additional families of bounds.Color energy graphs are the main focus of this paper, and the bounds on f ( n, p, q ) whichare proven using color energy graphs all fit into a general framework, which we describehere.Let F be a bipartite graph and r ≥ α > F is ( r, α ) -nice ifex( n, F ) = O ( n − α ) and f (cid:18) n, r | V ( F ) | , (cid:18) r | V ( F ) | (cid:19) − ( r − | E ( F ) | + 1 (cid:19) = Ω (cid:16) n αrr − (cid:17) . (2)Furthermore, we say that F is simply r -nice if F is ( r, α )-nice for every α > n, F ) = O ( n − α ). Intuitively, a graph F is r -nice if any coloring of K n with significantlyfewer than n αrr − colors has a clique of size r | V ( F ) | spanning as few colors as we would expectif there were r pairwise vertex-disjoint copies of F each with the same coloring.In [9], the authors used the color energy graph to prove that even cycles C k are 2-nice, whichled them to find new bounds on f ( n, p, q ) based on known upper bounds on ex( n, C k ). Theyalso showed that C is 3-nice, but did not prove that any infinite families of graphs are r -nice for any r ≥
3. We extend the techniques used in [9] to find many families of ( r, α )-nicegraphs. For t ≥
3, denote by K + t the subdivision of K t and by Θ ( a, b ) the theta graph ,which consists of b internally-disjoint paths of a edges each with the same two endpoints.Furthermore, for a ≤ b , let K ℓa,b denote the graph obtained from K a,b by replacing each edgewith a path of length ℓ (that is, K a,b = K a,b ). Theorem 1.2.
We have the following:(i) K + t is -nice for all t ≥ . (Theorem 2.3)(ii) Θ ( a, b ) is r -nice for all r, b ≥ , a > r . (Theorem 4.1)(iii) K ℓ ,b is ( r, − ℓ ) -nice for b ≥ , ℓ ≥ , r ∈ { , , , , } with r < ℓ . (Theorem 4.2) Each of these families gives us new lower bounds on f ( n, p, q ) for the appropriate choicesof p and q , many of which improve on existing bounds in the literature. For a detailedcomparison of our bounds to previous results, see Section 1.2.The fact that Θ ( a,
2) = C a is r -nice for a > r is particularly interesting due to the connectionwith the long-standing question about lower bounds for the extremal number ex( n, C a ). Thisextremal number is only known up to a multiplicative constant factor when a ∈ { , , } . Ifone could show that f (cid:18) n, r | V ( C k ) | , (cid:18) r | V ( C k ) | (cid:19) − ( r − | E ( C k ) | + 1 (cid:19) = O (cid:16) n αrr − (cid:17) , (3)for some α > r > a , then Theorem 4.1 would imply that ex( n, C a ) = Ω( n − α ). Inparticular, proving (3) for α = 1 − a would give a lower bound on ex( n, C a ) matching the3pper bound up to a multiplicative constant factor. It is unclear, however, if determiningthe value of f ( n, p, q ) for this specific choice of parameters would be easier than proving theextremal number for even cycles directly.It is worth noting that Theorem 1.1 applied with m = r −
1, and k = rt for some t impliesthat trees T with | V ( T ) | = t are r -nice, as it is well-known that ex( n, T ) = Θ( n ).Using a simple induction argument employed by Erd˝os and Gy´arf´as [6] in their originalpaper, we improve a result of Fish, Pohoata, and Sheffer [9] and obtain a generalization ofthe lower bound from [6] on f ( n, p, p ). Theorem 1.3.
For every ≤ m ≤ k − , we have f (cid:18) n, k, (cid:18) k (cid:19) − m ( k − m ) − (cid:18) m (cid:19) + m + 1 (cid:19) = Ω (cid:0) n /m (cid:1) . Moving forward, when considering f ( n, p, q ), instead of viewing the problem as requiring atleast q colors on every p -clique, it is often helpful to think about the problem in terms ofhaving at most (cid:0) p (cid:1) − q repetitions among existing colors. More formally, if p is an edge-colored clique and C is the set of all colors that appear on edges of p , we say that the clique p has (cid:0) p (cid:1) − | C | color repetitions or just repetitions . For accounting purposes, it will oftenbe useful to count repetitions in groups. For example, if we consider a set of k edges of thesame color c in a p -clique K , we will count this as k − c , in which case we will consider these k edges to haveyielded k repetitions. Motivated by the treatment of the Erd˝os-Gy´arf´as problem in [14], Conlon and Tyomkyn [3]asked how many colors are necessary in a proper edge-coloring of K n without many vertex-disjoint color-isomorphic copies of some fixed small graph H . (We say edge-colored graphsare color-isomorphic if there is an isomorphism between them preserving the colors.) Moreprecisely, for n, k ≥ H , we denote by f k ( n, H ) the smallest integer C such thatthere is a proper edge-coloring of K n with C colors containing no k disjoint color-isomorphiccopies of H . They consider only proper colorings to avoid some trivial obstacles, but theconnection to determining f ( n, p, q ) can be made explicit.Fix n, k ≥ H such that ( k − | E ( H ) | ≤ k | V ( H ) | −
2. Let p = k | V ( H ) | and q = (cid:0) k | V ( H ) | (cid:1) − ( k − | E ( H ) | + 1, and consider a ( p, q )-coloring of K n . While this coloringis not necessarily proper, it forbids monochromatic stars on p vertices, since such a starwould contain k | V ( H ) | − p, q )-coloring.Since each vertex is incident to a bounded number of edges of each color, we can obtaina proper coloring by expanding our set of colors by a constant factor. This new coloringcannot contain k disjoint color-isomorphic copies of H , otherwise we can find a p -clique4n the original coloring with fewer than q colors. Therefore, our ( p, q )-coloring must useΩ( f k ( n, H )) colors, and hence, we have f (cid:18) n, k | V ( H ) | , (cid:18) k | V ( H ) | (cid:19) − ( k − | E ( H ) | + 1 (cid:19) = Ω( f k ( n, H )) . (4)By exploiting this relationship between the two problems, we can obtain bounds on theErd˝os-Gy´arf´as function f ( n, p, q ) using known results about f k ( n, H ). For example, Conlonand Tyomkyn [3] gave a short proof that for every integer k and tree T with m edges, f k ( n, T ) = Ω( n /m ), which allows us to recover Theorem 1.1 without the need to invokecolor energy. More recently, Xu and Ge [18] showed that for t ≥ f ( n, K + t ) = Ω( n t − ).Applying (4) gives a result which matches Theorem 2.3 due to our current knowledge of theextremal number of K + t . Among other impressive results, Janzer [12] showed that for fixedintegers k, r ≥ f r ( n, C k ) = Ω (cid:16) n rr − · k − k (cid:17) . His proof can be extended in a straightforward manner to show f r ( n, Θ ( a, b )) = Ω (cid:16) n rr − · a − a (cid:17) , which matches our result in Theorem 4.1. In addition to giving non-trivial lower bounds on f ( n, p, q ) for new families of pairs ( p, q ),our work improves existing bounds for previously studied families of pairs ( p, q ). To see howour theorems improve existing bounds, note that f ( n, p, q ) ≤ f ( n, p, q ′ ) for q ≤ q ′ . By fixinga number of vertices p and a total number of colors, we can compare results by consideringthe number of repetitions that we can guarantee on each p -clique.Setting m = 2 t − k = 2 s in Theorem 1.1 [15] gives f (cid:18) n, s, (cid:18) s (cid:19) − (cid:18) t − t + 32 (cid:19) + 1 (cid:19) = Ω (cid:16) n t − (cid:17) , which we improve in Theorem 2.3, showing f (cid:18) n, s, (cid:18) s (cid:19) − ( t − t ) + 1 (cid:19) = Ω (cid:16) n t − (cid:17) . We can perform a similar comparison between Theorems 4.2 and 1.1 by considering the casewhen the number of colors n /m satisfies m = 3( r − ℓ ℓ − r r < ℓ . In this case, when r ≤ ℓ , our theorem gives the same number of repetitionson k = r (3 + (3 ℓ − b ) vertices as Theorem 1.1. However, when r > ℓ , Theorem 1.1gives f (cid:18) n, k, (cid:18) k (cid:19) − r − ℓb + m + 1 (cid:19) = Ω (cid:0) n /m (cid:1) while our Theorem 4.2 improves this to f (cid:18) n, k, (cid:18) k (cid:19) − r − ℓb + 1 (cid:19) = Ω (cid:0) n /m (cid:1) . Similarly, Theorem 1.3 improves the following result of Fish, Pohoata, and Sheffer [9], provedusing an application of the well-known K˝ov´ari-S´os-Tur´an Theorem [13].
Theorem 1.4 ([9]) . For any integers ≤ m ≤ k/ , f (cid:18) n, k, (cid:18) k (cid:19) − m ( k − m ) + 2 (cid:19) = Ω (cid:0) n /m (cid:1) . For comparison, in Theorem 1.3, we obtain the same bound for a smaller number of colorson each clique: f (cid:18) n, k, (cid:18) k (cid:19) − m ( k − m ) − (cid:18) m (cid:19) + m + 1 (cid:19) = Ω (cid:0) n /m (cid:1) . Proof of Theorem 1.3.
Suppose we color K n with c := n /m / (2( k − m ) /m ) = Θ( n /m ) colors.Arbitrarily choose a vertex v , and note that there exists a color such that v is incident withat least n − n /m / (2( k − m ) /m ) > ( k − m ) /m n − /m edges of this color, say color 1. Restricting to the neighborhood of v in color 1, arbitrarilychoose a vertex v , and we can find a color such that v is incident with at least( k − m ) /m n − /m − n /m / (2( k − m ) /m ) > ( k − m ) /m n − /m edges of this color, say color 2, all of whose endpoints are in the color 1 neighborhood of v .Continue iteratively, until we have selected vertices v , v , . . . , v m and colors 1 , , . . . , m suchthat there at least k − m = ( k − m ) m/m n − m/m vertices simultaneously in the i -th color-neighborhood of v i for all 1 ≤ i ≤ m . Then we havea set of k vertices that spans at most m + (cid:18) k − m (cid:19) = (cid:18) k (cid:19) − m ( k − m ) − (cid:18) m (cid:19) + m colors. Thus, any coloring which hopes to have every clique span one more color needs atleast Ω( c ) = Ω( n /m ) colors, completing the proof. (cid:4) r, b ≥ a > r , for ℓ = 2 + b ( a − C, c > n sufficientlylarge, cn rr − · a − a ≤ f (cid:18) n, rℓ, (cid:18) rℓ (cid:19) − ( r − ab + 1 (cid:19) ≤ Cn rr − · a − a + ab . Note that as b increases, the gap between the lower and upper bounds shrinks (although C and c implicitly depend on b ). The rest of the paper is organized as follows. The proofs of Theorems 2.3, 4.1, and 4.2 increasein difficulty, so as we develop the concept of the color energy graph and associated tools, weprove these theorems when we have sufficient techniques to do so. In Section 2, we define thecolor energy graph and a helpful variant called the pruned color energy graph (Section 2.1).These tools are sufficient to provide a short proof of Theorem 2.3 in Section 2.2. In Section 3,we develop terminology and theory necessary for finding more complicated structures withthe color energy graph. Following this development, we prove Theorem 4.1 in Section 4.1.Finally, in Section 4.2, we give the more involved proof of Theorem 4.2. We provide someavenues for future research in Section 5.
In analogy to the additive energy of additive combinatorics, Pohoata and Sheffer [15] de-fined color energy of an edge-colored graph. With Fish [9], they went further in defining acorresponding graph, the “color energy graph.” As we use this graph extensively, we collecthere its definition, some basic results, and a few helpful modifications to the color energygraph.
Definition.
Given a graph G = ( V, E ) with a coloring χ : E → C , the r -th color energygraph ~G = ( ~V , ~E ) has vertex set V r with an edge between ( v , . . . , v r ) and ( u , . . . , u r ) if andonly if χ ( v u ) = . . . = χ ( v r u r ).If it is clear from context, we will omit the r -th in the name and simply refer to ~G as the colorenergy graph. Note that ~V includes r -tuples with repeated coordinates, but ~G is looplessas G is loopless. Since an edge of the color energy graph corresponds to a multiset of r edges of the same color in G , the coloring χ also naturally extends to a coloring on ~G . Thefollowing relation between | E | , | ~E | , and | C | allowed the authors in [9] to derive lower boundson f ( n, p, q ). Fish, Pohoata, and Sheffer [9] defined the r -th color energy graph as a subgraph of ~G obtained byremoving certain edges such as loops and edges of “unpopular” colors. For clarity, we instead gather thenecessary restrictions on ~E for our proofs in the definition of a pruned energy graph in Section 2.1. roposition 2.1 ([9]) . If G = ( V, E ) with coloring χ : E → C has r -th color energy graph ~G = ( ~V , ~E ) , then | C | ≥ | E | r | ~E | ! r − . Proof.
For each color c ∈ C , let m c be the number of edges of color c in G . Observe that P c ∈ C m c = | E | and P c ∈ C m rc = | ~E | . H¨older’s inequality implies | ~E | = X c ∈ C m rc ≥ (cid:0)P c ∈ C m c (cid:1) r (cid:0)P c ∈ C (cid:1) r − = | E | r | C | r − . (cid:4) We fix some terminology here. Let G and H be graphs. A graph homomorphism from H to G is a function φ : V ( H ) → V ( G ) such that if uv ∈ E ( H ) then φ ( u ) φ ( v ) ∈ E ( G ). Sucha function on V ( H ) induces a function on E ( H ) which we also call φ . As a graph, φ ( H ) iscalled a homomorphic image (or just image ) of H in G . If φ is injective, then φ ( H ) is calledan isomorphic copy (or just copy ) of H in G . If the edges of H are colored and φ ( H ) inheritsthis coloring, then φ ( H ) is called a color-homomorphic image (likewise color-isomorphiccopy ) of H in G .All of the proofs using the color energy graph follow roughly the same format. We fix agraph H , start with an ( r | V ( H ) | , (cid:0) r | V ( H ) | (cid:1) − ( r − | E ( H ) | + 1)-coloring χ of a completegraph K n = ( V, E ), and consider the r -th color energy graph ~G = ( ~V , ~E ). Let π k : ~V → V be the k -th coordinate map for 1 ≤ k ≤ r . If a copy ~H of H is found as a subgraph of ~G , then π k ( ~H ) is a color-homomorphic image of ~H for each k ∈ [ r ]. If these r color-homomorphicimages of ~H are actually pairwise vertex-disjoint color-isomorphic copies of ~H , then takentogether, these copies span r | V ( H ) | vertices and contain at least ( r − | E ( H ) | repetitions,contradicting that χ is an ( r | V ( H ) | , (cid:0) r | V ( H ) | (cid:1) − ( r − | E ( H ) | + 1)-coloring. In this case, H is not a subgraph of ~G , so we have an upper bound on | ~E | in terms of | ~V | via the extremalnumber of H . By Proposition 2.1, this gives a lower bound on the number of colors used by χ , and hence a lower bound on f ( n, p, q ).Unfortunately, it is not guaranteed that the images π k ( ~H ) are disjoint color-isomorphiccopies of ~H . This lead the authors in [9] to “prune” the color energy graph so that, forparticular H , the color-homomorphic images still contain a sufficient number of repetitionsand vertices. We concisely describe this pruning in the next subsection, as we use this forour proofs utilizing the color energy. Let K be a graph. We say that a homomorphism φ : V ( H ) → V ( G ) (and the homomorphicimage φ ( H )) is K -preserving if every copy of K in H is mapped to a copy of K in G under8 . We denote by P k the path on k vertices.We cannot guarantee that the existence of a graph ~H ⊆ ~G will yield k disjoint color-isomorphic copies of ~H , but we can guarantee that these color-homomorphic images of ~H are disjoint, bipartite, and P -preserving. Definition.
Let G = ( V, E ) be a graph with coloring χ : E → C . A pruned r -th energygraph is a subgraph ~G ′ = ( ~V ′ , ~E ′ ) of the r -th color energy graph ~G = ( ~V , ~E ) with thefollowing structure:1. There is a partition V = V ∪ · · · ∪ V r such that ~V ′ = V × · · · × V r and ⌊| V | /r ⌋ ≤ | V i | ≤⌈| V | /r ⌉ for each i .2. For every i , there exist partitions V i = V ′ i ∪ V ′′ i such that the i -th coordinate of everyedge of ~E ′ has one endpoint in each of V ′ i and V ′′ i .3. If ~x, ~y ∈ ~V ′ are at distance at most 2 in ~G ′ , then ~x and ~y are not equal in any coordinate.We now fix some notation. In general, the vertices of a color energy graph are denoted witha vector arrow above them, as in ~v , to remind the reader that they are tuples of vertices of G . We speak of entries in the tuples as ‘coordinates.’ We let π k : V ( ~G ) → V k be the k -thcoordinate map, which induces a map on the edges of ~G as well. We abuse notation and alsocall this edge map π k . Note that by property 3, edges in the pruned energy graph ~G ′ are sentto edges in G under π k . Furthermore, we let π : ~G ′ → G be defined by π ( ~H ) = S k π k ( ~H ).We often consider paths in ~G ′ . By property 3, the image of a path under π k is a walk whichmay repeat edges and vertices, but will never repeat the same edge consecutively, that is, itwill never ‘turn around.’The utility of the pruned energy graph is that the additional conditions come at no costto the growth rate of the number of edges, as long as we consider colorings in which everyvertex is incident to a bounded number of edges of each color. To enforce this color degreecondition, we will require all our colorings to be ( p, (cid:0) p (cid:1) − p + 3)-colorings, which forbidsmonochromatic stars on p vertices. Unfortunately, this implies that we can only use apruned color energy graph when we want to prove bounds involving a superlinear number ofcolors since ( p, (cid:0) p (cid:1) − p + 3)-colorings of K n require Ω( n ) colors. The existence of a prunedenergy graph was shown in [9], so we only give a sketch of the proof below. Proposition 2.2 ([9]) . Let G = ( V, E ) be a graph with a ( p, (cid:0) p (cid:1) − p + 3) -coloring χ : E → C ,and let ~G = ( ~V , ~E ) be the r -th color energy graph. Then, there exists a pruned r -th energygraph ~G ′ = ( ~V ′ , ~E ′ ) such that | ~V ′ | = Θ( | ~V | ) and | ~E ′ | = Θ( | ~E | ) .Proof Sketch. A standard probabilistic argument shows that every graph has a bipartitesubgraph with at least half as many edges as the original. To get the partition V , . . . , V r ,one can use a modification of this argument to ensure that a constant fraction of the edges of ~G have all coordinates with both endpoints in the same V i . One achieves property 2 througha similar probabilistic argument. 9or property 3, we need that every vertex is incident to at most p − χ is a ( p, (cid:0) p (cid:1) − p + 3)-coloring). Among the neighbors of a vertex ~z withfirst coordinate v , there are at most p − π ( ~z ) isincident to at least p − χ ( π ( ~z ) v ). Thus, ~z has at most ( p − r − neighborswith first coordinate v . We keep one of them for each ~z and v , giving property 3 for verticesat distance 2. Since G is loopless, property 3 also holds for vertices at distance 1. (cid:4) In this section, we show that K + t is 2-nice. Using the boundex( n, K + t ) = O (cid:16) n − t − (cid:17) given by Janzer [10] for t ≥
3, we obtain the following result.
Theorem 2.3.
Let t ≥ . Then K + t is -nice. Consequently, if s := | V ( K + t ) | = t + (cid:0) t (cid:1) (note that | E ( K + t ) | = 2 (cid:0) t (cid:1) ) , then we have f (cid:18) n, s, (cid:18) s (cid:19) − (cid:18) t (cid:19) + 1 (cid:19) = Ω (cid:16) n t − (cid:17) . Proof.
Let α > n, K + t ) = O ( n − α ), and let G = ( V, E ) be a completegraph K n , C be a set of colors, and χ : E → C be a (2 s, (cid:0) s (cid:1) − (cid:0) t (cid:1) + 1)-coloring of G , where s = | V ( K + t ) | = t + (cid:0) t (cid:1) . Consider the pruned 2nd energy graph ~G = ( ~V , ~E ). Assume thatthere is a copy ~K of K + t in ~G with vertices ~x , . . . , ~x t and ~y , . . . , ~y ( t ). Recall that π k ( ~K )is P -preserving and bipartite for k = 1 ,
2. Then clearly no ~x i and ~y j share any commoncoordinates, because of the bipartite structure of ~G , and all of the coordinates of ~x , . . . , ~x t are distinct by the P -preserving property. While it is possible for the same coordinate toappear in multiple vertices ~y , . . . , ~y ( t ), corresponding to “degenerate” homomorphisms of K + t in G , this will not change the number of distinct edges that we find in the two copiesof K + t in G . (If it did, then it would happen because two vertices ~y i and ~y j adjacent tothe same vertex ~x k shared a coordinate, which is forbidden in a pruned energy graph.) So, π k ( ~K ) has 2 (cid:0) t (cid:1) edges for k = 1 ,
2, giving us a clique on at most 2 s vertices with 2 (cid:0) t (cid:1) colorrepetitions, contradicting the assumption about our coloring. Therefore, we have | E ( ~G ) | ≤ ex( n , K + t ) = O (cid:0) n − α (cid:1) , and hence by Propositions 2.2 and 2.1, | C | = Ω ( n α ). Thus, K + t is 2-nice, and letting α = 1 / / (4 t −
6) yields the result. (cid:4)
In the above proof we only considered the 2nd energy graph. That is because in order forthe r -th pruned energy graph to exist, by Proposition 2.2, we need r | V ( K + t ) | − ≥ ( r − | E ( K + t ) | − , (5)10s otherwise we cannot guarantee that the color degrees of each vertex are bounded. In-equality (5) is only true for all t when r = 2. For larger r , some values of t still satisfy (5),but these choices of t either do not give new bounds or are covered elsewhere. For example,setting r = 3 and t = 3 gives K +3 = C , which is covered in Theorem 4.1.The proof of Theorem 2.3 was relatively straightforward because P -preserving homomor-phisms of K + t are easy to understand: all the vertices corresponding to the original K t mustbe distinct, and the “subdivision” vertices only coincide when the edges of the original K t that correspond to these vertices form a matching. In particular, all P -preserving homo-morphisms of K + t have the same number of edges, which was useful in the proof. For otherstructures, P -preserving homomorphisms are not so easily understood, and may not havethe same number of edges as the original graph. In the next section, we develop tools to helpus analyze P -preserving homomorphic images of general graphs, allowing us to apply thecolor energy techniques to theta graphs and subdivided complete bipartite graphs (wherethe edges are subdivided an arbitrary number of times). In this section, we develop a framework that will help us prove Theorems 4.1 and 4.2, andalso may be useful for proving further results outside the scope of this paper.As usual, let G be an edge-colored graph and ~G be a pruned r -th energy graph of G . Let H ⊆ G and ~T ⊆ ~G with m := | E ( ~T ) | . We denote by H k = H ∩ G [ V k ] the k -th coordinate of H , for each 1 ≤ k ≤ r . An ordering σ of E ( ~T ) = { ~e , . . . , ~e m } is called H -compatible if forevery i ∈ [ m ], there exists an endpoint ~v of ~e i such that π ( ~v ) ⊆ H ∪ i − [ j =1 π ( ~e j ) . Given graphs H ⊆ G and ~T ⊆ ~G , we wish to understand the number of vertices and thenumber of repetitions in H ∪ π ( ~T ). To do this, we give a simple algorithm that adds verticesfrom π ( ~T ) to H in | E ( ~T ) | steps, given an H -compatible ordering of E ( ~T ). During thealgorithm we keep track of several parameters, which in the proofs of Theorems 4.1 and4.2 allow us to leverage the structure of ~T and H to analyze the number of vertices andrepetitions π ( ~T ) adds to H .Let m := | E ( ~T ) | , and let E ( ~T ) = { ~e , . . . , ~e m } , where the ordering of the edges, say σ , is H -compatible. Let ~e i = ~u i ~v i , where π ( ~u i ) ⊆ H ∪ S i − j =1 π ( ~e j ). We recursively build graphs H = H (0) , H (1) , . . . , H ( m ) = H ∪ π ( ~T ), where for 1 ≤ i ≤ m , H ( i ) = H ( i − ∪ π ( ~e i ) .
11e sequentially add the edges of ~T to H according to the H -compatible ordering of E ( ~T ).We now define some terminology which will be useful for analyzing this process. • We say that step i gives us a new vertex in coordinate k if the vertex π k ( ~v i ) V ( H ( i − k )(which implies that the edge π k ( ~e i ) E ( H ( i − k )). Let n i,k ( ~T , H, σ ) := 1 if we get anew vertex in coordinate k on step i , and n i,k ( ~T , H, σ ) := 0 otherwise. • We say that step i gives us a savings in coordinate k if the vertex π k ( ~v i ) ∈ V ( H ( i − k ),but the edge π k ( ~e i ) E ( H ( i − k ). Let s i,k ( ~T , H, σ ) := 1 if we get a savings in coordinate k on step i , and s i,k ( ~T , H, σ ) := 0 if we do not. • We say that step i gives us a delayed vertex in coordinate k if the edge π k ( ~e i ) ∈ E ( H ( i − k ) (which implies that the vertex π k ( ~v i ) ∈ V ( H ( i − k )). Set d i,k ( ~T , H, σ ) := 1 ifwe get a delayed vertex in coordinate k on step i , and d i,k ( ~T , H, σ ) := 0 otherwise.Often ~T , H and σ will be clear from context. In those cases, we will omit them from thenotation. By definition, n i,k + s i,k + d i,k = 1. We now define aggregate parameters derivedfrom n i,k , s i,k and d i,k . Let n i := r X k =1 n i,k , N := m X i =1 n i , N k := m X i =1 n i,k ,s i := r X k =1 s i,k , S := m X i =1 s i , S k := m X i =1 s i,k ,d i := r X k =1 d i,k , D := m X i =1 d i , D k := m X i =1 d i,k be the new vertices/savings/delayed vertices in step i , in total, and in each coordinate,respectively. We emphasize here that each of these parameters are functions of ~T , H and σ ,so we may write N ( ~T , H, σ ), S k ( ~T , H, σ ) or other parameters with these variables in caseswhere the triple ( ~T , H, σ ) is not clear from context. Summing n i,k + s i,k + d i,k = 1 over k ,we get for all 1 ≤ i ≤ m n i + s i + d i = r, (6)and summing this over i , we get N + S + D = rm. Finally, let d := m X i =1 ( d i =0) be the number of steps where there are no delayed vertices.12e can precisely describe the number of vertices and repetitions added to H by ~T under theordering σ using these parameters. By the definition of N , we have | V ( H ∪ π ( ~T )) | = | V ( H ) | + N = | V ( H ) | + rm − S − D. (7)Furthermore, if R is the number of repetitions in H and R ∗ is the number of repetitions in H ∪ π ( ~T ), then R ∗ − R ≥ m X i =1 (cid:0) n i + s i − ( d i =0) (cid:1) = rm − D − d. (8)Indeed, in step i , we introduce exactly n i + s i new edges, all of the same color, say c . If d i = 0, then this constitutes r new edges of the same color, givings us r − n i + s i − d i = 0, then for some k , π k ( ~u i ~v i ) ∈ E ( H ( i − k ), which implies that therealready was an edge of color c present in H ( i − k , so all n i + s i new edges are repetitions. After performing the graph revealing algorithm, if there are many delayed vertices, then wedo not expect to have as many repetitions as we desire. However, we also do not have asmany vertices in H ∪ π ( ~T ) as we expected, and to capitalize on that, we wish to add morevertices to get more repetitions. In general, for every r new vertices, we expect r new edges,and thus r − D delayed vertices, we wish to get about r − r D extra repetitions. The goal of this subsection is to make this more precise. We achieve theseextra repetitions via an easy-to-find gadget in ~G .Let H ⊆ G . An H -reservoir with source ~v is a set ~R ⊆ V ( ~G ) along with a vertex ~v ∈ V ( ~G )such that the following holds:1. π ( ~v ) ⊆ H ,2. ~u~v ∈ E ( ~G ) for all ~u ∈ ~R ,3. H and π ( ~u ) are disjoint for all ~u ∈ ~R .Note that if π ( ~v ) ⊆ F ⊆ H and ~R is an H -reservoir with source ~v , then ~R is also an F -reservoir. We now show that H -reservoirs allow us to add repetitions we missed from delayedvertices. Lemma 3.1.
Let H ⊆ G , and let ~R be an H -reservoir. For any non-negative integer D with D ≤ r | ~R | there exists some graph H ∗ ⊆ G that satisfies the following: • H ⊆ H ∗ ⊆ H ∪ π ( ~R ) , • | V ( H ∗ ) | = | V ( H ) | + D , and • H ∗ has at least j ( r − r D k more repetitions than H . roof. Let ~v be the source of the reservoir ~R . Let w, z be integers with 0 ≤ z < r such that D = wr + z , and note that (cid:22) ( r − r D (cid:23) = ( r − w + z − ( z =0) . Choose w vertices from ~R , say ~v , . . . , ~v w , and add π ( ~v~v i ) to H to form H ′ . Note that since ~v ~v i ∈ E ( ~G ) for all 1 ≤ i ≤ w , there are r edges in E ( H ′ ) \ E ( H ) from vertices in π ( ~v ) tovertices in π ( ~v i ), all of which are the same color, giving us ( r − w new repetitions. If z = 0,then H ′ satisfies the statement of the lemma, so we are done. If z = 0, then let ~v w +1 be anyvertex in ~R \ { ~v , . . . , ~v w } . Add π k ( ~v w +1 ) to H ′ for 1 ≤ k ≤ z to form H ∗ . Note that thisgives us a collection of z more edges, all of the same color, yielding z − H ∗ satisfies the conditions of the lemma. (cid:4) Theorem 3.2.
Let H ⊆ G and ~T ⊆ ~G be graphs. Fix some H -compatible ordering of E ( ~T ) ,and let ~R be an H ∪ π ( ~T ) -reservoir with | ~R | ≥ ⌈ D/r ⌉ . Let m := | E ( ~T ) | . If S + m X i =1 ( d i > (cid:18) r − d i r − (cid:19) ≥ t, (9) for some t ≥ , then there exists a graph H ∗ with H ∪ π ( ~T ) ⊆ H ∗ ⊆ H ∪ π ( ~T ) ∪ π ( ~R ) , suchthat | V ( H ∗ ) | ≤ | V ( H ) | + rm − t, and the number of repetitions in H ∗ that are not in H is at least ( r − m. Proof.
Let H ′ = H ∪ π ( ~T ). By (7), | V ( H ′ ) | = | V ( H ) | + rm − S − D, and by (8), H ′ has rm − D − d more repetitions than H does. Let D ′ := S + D − t ≥ D − m X i =1 ( d i > (cid:18) r − d i r − (cid:19) . Apply Lemma 3.1 (with H ′ as H , ~R as ~R , and D ′ as D ) to find a graph H ∗ with H ′ ⊆ H ∗ ⊆ H ′ ∪ π ( ~R ) such that | V ( H ∗ ) | = | V ( H ′ ) | + D ′ = | V ( H ) | + rm − t, H ∗ has the correct number of vertices. In addition, H ∗ has j ( r − r D ′ k more repetitionsthan H ′ . This gives us that H ∗ has at least rm − D − d + (cid:22) r − r D ′ (cid:23) ≥ rm − D − d + $ r − r D − m X i =1 ( d i > (cid:18) r − d i r − (cid:19)!% = ( r − m + $ m − Dr − d − m X i =1 ( d i > (cid:18) r − d i r (cid:19)% = ( r − m + (cid:22) m − Dr − d − ( m − d ) + Dr (cid:23) = ( r − m repetitions that are not in H . (cid:4) In light of Theorem 3.2, given graphs H and ~T (with a fixed H -compatible ordering of E ( ~T )), we define the total savings of ~T with respect to H and the ordering σ , sav( ~T , H, σ ),to be sav( ~T , H, σ ) := S + m X i =1 ( d i > (cid:18) r − d i r − (cid:19) . In this subsection, we will provide some nice properties of the parameters we get from thegraph revealing algorithm.
Order-Invariance
First we show that given graphs H ⊆ G and ~T ⊆ ~G , many of the parameters given by thegraph revealing algorithm are constant over all H -compatible orderings of ~T . Observation 3.3.
Given graphs H ⊆ G and ~T ⊆ ~G , for every k ∈ [ r ] the quantities N k , S k and D k are constant across all H -compatible orderings of ~T . Consequently, the quantities N , S , and D are also constant across all H -compatible orderings of ~T .Proof. Let σ be an H -compatible ordering of ~T . Since n i,k ( ~T , H, σ ) + s i,k ( ~T , H, σ ) + d i,k ( ~T , H, σ ) = 1 , we have that N k + S k + D k = m , so it will suffice to show that two of these parameters areconstant (with respect to σ ). First consider N k , and note that N k ( ~T , H, σ ) = | V ( H ∪ π k ( ~T )) | − | V ( H ) | , σ . Now consider D k . Given anedge e ∈ E ( G ), let π − k ( e ) be the preimage of e under π k , or the set containing every edge of ~T that gets mapped to e . Then D k ( ~T , H, σ ) = X e ∈ E ( G ) max { , | π − k ( e ) | − ( e/ ∈ E ( H )) } . Indeed, regardless of the ordering on E ( ~T ), if e ∈ E ( H ), then every edge of π − k ( e ) gives usone delayed vertex in coordinate k on the step it is revealed. If e E ( H ), then the firstedge revealed in π − k ( e ) does not give us a delayed vertex in coordinate k , but all others do.Note again that the expression we derived for D k ( ~T , H, σ ) is independent of σ . (cid:4) It is worth noting that the parameter d is not necessarily constant across all H -compatibleorderings. In light of the preceding observation, in cases where we need to clarify H and ~T , we may write parameters such as S k ( ~T , H ), D ( ~T , H ), and others without reference to σ . Since the parameters n i,k , s i,k , d i,k , n i , s i , d i and d all require the edge-ordering σ to bewell-defined though, we will specify σ as appropriate. Additivity
Let ~T , ~T ⊆ ~G be edge-disjoint graphs and σ and σ be H -compatible orderings of ~T and ~T respectively. Set ~T := ~T ∪ ~T and let σ be the H -compatible ordering of ~T given by firstordering the edges of ~T in the order given by σ , then ordering the edges of ~T in the ordergiven by σ . Then we havesav( ~T , H, σ ) = sav( ~T , H, σ ) + sav( ~T , H ∪ π ( ~T ) , σ ) . (10) Monotonicity
The parameters given by the graph revealing process also satisfy certain monotonicity prop-erties with respect to subgraphs. More specifically, if F ⊆ F ⊆ G and ~T ⊆ ~G are graphs,and σ is an F -compatible ordering of ~T (and F -compatible since F ⊆ F ), then n i,k ( ~T , F , σ ) ≥ n i,k ( ~T , F , σ )and d i,k ( ~T , F , σ ) ≤ d i,k ( ~T , F , σ )for all 1 ≤ i ≤ | E ( ~T ) | and k ∈ [ r ]. Indeed, if n i,k ( ~T , F , σ ) = 1, then n i,k ( ~T , F , σ ) = 1, sincethe second endpoint of π k ( ~e i ) cannot be in F ∪ S i − j =1 π ( ~e j ) if it is not in F ∪ S i − j =1 π ( ~e j ).Similarly, if d i,k ( ~T , F , σ ) = 1, then d i,k ( ~T , F , σ ) = 1, since π k ( ~e i ) ⊆ F ∪ S i − j =1 π ( ~e j ) implies π k ( ~e i ) ⊆ F ∪ S i − j =1 π ( ~e j ) as well. Consequently, we have that N k ( ~T , F ) ≥ N k ( ~T , F ) (11)16nd D k ( ~T , F ) ≤ D k ( ~T , F ) . (12)Unfortunately, s i,k does not satisfy such a monotonicity property. In general, steps in whichwe get a vertex savings with respect to F may be new vertices with respect to F , and stepsin which we get delayed vertices with respect to F may be vertex savings with respect to F , so the number of vertex savings may increase or decrease when revealing ~T with respectto a subgraph. Properties of Revealing Paths
Often the graph ~T we reveal with the graph revealing algorithm is a path (or a collection ofpaths). Given a path ~P ⊆ ~G with endpoints ~u and ~v , the canonical ordering of ~P from ~u to ~v is the ordering σ of E ( ~P ) = { e , e , . . . , e ℓ } such that e i appears before e i +1 on ~P as wetraverse ~P from ~u to ~v . The following lemma will be very useful for finding vertex savingswhile revealing paths. Lemma 3.4.
Let F ⊆ G . Let ~P = ( ~v , ~v , . . . , ~v ℓ ) be a path in ~G such that π ( ~v ) ⊆ F , andlet E ( ~P ) = { ~e j | j ∈ [ ℓ ] } be canonically ordered from ~v to ~v ℓ . If there exists a choice of k ∈ [ r ] and j, j ′ ∈ [ ℓ ] with j ≤ j ′ such that1. π k ( ~e j ) E (cid:16) F ∪ S j − i ′ =1 π k ( ~e i ′ ) (cid:17) , and2. π k ( ~v j ′ ) ∈ V ( F ) ,then there is a vertex savings in coordinate k with respect to F as we reveal the path ~P atsome step j ∗ with j ≤ j ∗ ≤ j ′ .Proof. By (1.) above, step j is either a new vertex or a vertex savings in coordinate k . Ifstep j is a vertex savings, then we are done. If not, and step j is a new vertex, let j ∗ > j bethe first index such that π k ( ~v j ∗ ) ∈ V F ∪ j ∗ − [ i ′ =1 π k ( ~e i ′ ) ! . Note that such an index exists and j ∗ ≤ j ′ since π k ( ~v j ′ ) ∈ V ( F ). Then we claim step j ∗ isa vertex savings in coordinate k . Indeed, step j ∗ does not constitute a new vertex by thedefinition of j ∗ , and step j ∗ cannot be a delayed vertex since step j ∗ − F (and thus by the P -preserving property of π k , π k ( ~e j ∗ ) E ( F ∪ S j ∗ − i ′ =1 π k ( ~e i ′ )).Therefore, step j ∗ gives a vertex savings as claimed, completing the proof. (cid:4) Recall that Θ ( a, b ) consists of b internally-disjoint paths of a edges each with the same twoendpoints, and K ℓa,b is the graph obtained from K a,b by replacing each edge with a path17ith ℓ edges. In this section, we apply the techniques of Section 3 to show that Θ ( a, b )is ( r, a − a )-nice for all a > r ≥ b ≥
2, and that K ℓ ,b is ( r, − ℓ )-nice for b, ℓ ≥
3, and3 ≤ r ≤ r < ℓ . For the following theorem, we will use the fact that for a, b ≥ n, Θ ( a, b )) = O ( n /a ) , and that this result is tight when b ≥ Theorem 4.1.
Let a > r ≥ and b ≥ . Then Θ ( a, b ) is r -nice. Consequently, letting ℓ := 2 + b ( a −
1) = | V ( Θ ( a, b )) | , we have f (cid:18) n, rℓ, (cid:18) rℓ (cid:19) − ( r − ab + 1 (cid:19) = Ω (cid:16) n rr − · a − a (cid:17) . Proof.
Let G = ( V, E ) be a complete graph K n , let C be a set of colors, and let χ : E → C bean ( rℓ, (cid:0) rℓ (cid:1) − ( r − ab +1)-coloring of G , where ℓ = 2+ b ( a − χ is an ( rℓ, (cid:0) rℓ (cid:1) − rℓ +3)-coloring, by Proposition 2.2 there exists a pruned r -th energy graph ~G = ( ~V , ~E ) of G .We address the cases b = 2 and b ≥ b ≥
3, we know thatex( n, Θ ( a, b )) = Θ( n /a ). For b ≥
3, we actually find Θ ( a, b ′ ) in the pruned energy graph,where b ′ is much larger than b , and use this Θ ( a, b ′ ) to find a copy of Θ ( a, b ) with a reservoir.For b = 2, Θ ( a, b ) = C a , and the extremal number ex( n, C a ) is only known (up to constantfactors) when a = 2, 3, or 5. Thus for b = 2, we must find the reservoir differently than for b ≥ Case 1: b = 2. Let α > n, C a ) = O ( n − α ). Note that α ≤ C a ∪ S k be the graph on 2 a + k vertices formed from the cycle C a by adding k degree1 vertices all adjacent to the same vertex in the cycle. We claim that for constant k ,ex( n, C a ∪ S k ) ≤ ex( n, C a ) + 2 a (2 a + k ) n = Θ(ex( n, C a )) . Indeed, a graph on n vertices with ex( n, C a ) + 2 a (2 a + k ) n edges must contain at least2 a (2 a + k ) n/ a = (2 a + k ) n edge-disjoint copies of C a , and so there must be a vertex v in 2 a + k copies. This implies that v is in a copy of C a and has degree at least 2 a + k , soeven if v has 2 a − k neighbors outside of the cycle,forming a copy of C a ∪ S k .Now, suppose for the sake of contradiction that ~G contains a copy ~C of C a ∪ S a ( r +1) . Let ~v denote the vertex of degree 2 a ( r + 1) + 2 in ~C , and ~u denote a neighbor of ~v in ~C of degree2 in ~C . Let ~X be the set of degree 1 vertices in ~C . We will let H = π ( ~u~v ), and ~T be the ~u – ~v -path of length 2 a − ~C . Let ~R consist of those vertices of ~X whose coordinates aredisjoint from H ∪ π ( ~T ), so that ~R is a H ∪ π ( ~T )-reservoir. We claim that | ~R | ≥ a . Note18hat there are at most 2 ar vertices in H ∪ π ( ~T ). Since all vertices in ~X are distance at mosttwo from each other, each vertex in H ∪ π ( ~T ) is a coordinate in at most one vertex in ~X .This forbids at most 2 ar vertices of ~X from being in ~R , and hence | ~R | ≥ a .Now, let us reveal ~T , where E ( ~T ) is canonically ordered from ~u to ~v . We claim that S ( ~T , H ) ≥ r . Indeed, by the P -preserving property, the first edge revealed constitutes a new vertex inall r coordinates, and then by Lemma 3.4, since the endpoint π k ( ~v ) of π k ( ~T ) is in V ( H ) foreach k ∈ [ r ], we get at least one savings in each coordinate, giving us r savings altogether.Therefore, by Theorem 3.2 applied with t = r , there is a graph H ∗ ⊆ G with | V ( H ∗ ) | ≤ | V ( H ) | + r | E ( ~T ) | − r ≤ r + r (2 a − − r = 2 ar, and at least ( r − | E ( ~T ) | = 2 a ( r − H ∗ , contradicting the choice of coloring on G . Thus, ~G does not containa copy of C a ∪ S a ( r +1) , so | E ( ~G ) | ≤ ex( n r , C a ∪ S a ( r +1) ) = O (cid:0) n r (2 − α ) (cid:1) . By Propositions 2.1 and 2.2, this implies | C | = Ω (cid:16) n αrr − (cid:17) . This means that C a is ( r, α )-nicefor any α , and thus C a is r -nice. Case 2: b ≥
3. Since ex( n, Θ ( a, b )) = Θ( n /a ), to show that Θ ( a, b ) is r -nice, it suffices toshow that Θ ( a, b ) is ( r, − /a )-nice. Suppose for the sake of contradiction that ~G containsa copy of Θ := Θ ( a, ra b ). Let ~v , ~v a ∈ ~V denote the vertices of degree greater than 2 in Θ .Label the paths ~P i = ( ~v = ~v i, , ~v i, , . . . , ~v i,a − , ~v i,a = ~v a ) for 1 ≤ i ≤ ra b . We may select asequence of distinct paths ~P i , . . . , ~P i ab + b such that ~v i j , has no coordinates in common withany vertices in { ~v , ~v a } ∪ S j − k =1 V ( ~P i k ) for all 1 ≤ j ≤ ab + b . This is possible because thereare at most ( a − j −
1) + 2 ≤ ( a − ab + b ) + 2 ≤ a b vertices in this union, creatingat most 2 ra b possible coordinate conflicts with possible choices for ~v i j , . Since the vertices ~v i, for 1 ≤ i ≤ ra b are all distance 2 from one another, our pruning guarantees that eachpossible coordinate conflict eliminates at most one choice of ~v i, for ~v i j , . Thus, there existssuch a choice for i , . . . , i ab + b . We reorder the paths so that these paths come first (that is, i j = j for all 1 ≤ j ≤ ab + b ), and we discard the remaining paths.Now let ~T = S bi =1 ~P i , H = π ( ~v ) ∪ π ( ~v a ), and ~R = { ~v i, | b + 1 ≤ i ≤ b + ab } . Let σ be anordering of E ( ~T ) such that the edges of ~P i appear before the edges of ~P j for all 1 ≤ i < j ≤ b ,and within each path, the edges are given the canonical ordering from endpoints ~v to ~v a .By the ordering placed on the paths in Θ , the vertices in ~R do not have any coordinates incommon with each other or any vertices in ~T , so ~R is a H ∪ π ( ~T )-reservoir with source ~v .We claim that S ( ~T , H ) ≥ rb . In fact, each path ~P i gives us r vertex savings when revealedwith respect to H ∪ S i − j =1 π ( ~P j ). Indeed, since π k ( ~v a ) ∈ H and since the edge π k ( ~v ~v i, ) ∪ S i − j =1 π ( ~P j ) by the ordering placed on the paths, Lemma 3.4 implies that there is a vertexsavings in coordinate k as we reveal ~P i for each k ∈ [ r ]. Thussav( ~T , H, σ ) ≥ rb. Therefore, by Theorem 3.2 applied with t = rb , there is a graph H ∗ ⊆ G with | V ( H ∗ ) | ≤ | V ( H ) | + r | E ( ~T ) | − rb = 2 r + rab − rb = rℓ and at least ( r − | E ( ~T ) | = ( r − ab repetitions. This contradicts the choice of coloring on G , so ~G does not contain a copy of Θ ( a, ra b ). Thus, | E ( ~G ) | ≤ ex( n r , Θ ( a, ra b )) = O (cid:0) n r + ra (cid:1) and hence by Propositions 2.1 and 2.2, | C | = Ω (cid:16) n rr − · a − a (cid:17) . (cid:4) To prove the following theorem, we will use the bound ex( n, K ℓa,b ) = O ( n a − aℓ ) given byJanzer [11]. Theorem 4.2.
Let b ≥ , ℓ ≥ , and ≤ r ≤ with r < ℓ . Then K ℓ ,b is ( r, − ℓ ) -nice.In other words, letting s := | V ( K ℓ ,b ) | = 3 + b + 3( ℓ − b and noting that | E ( K ℓ ,b ) | = 3 bℓ , wehave f (cid:18) n, rs, (cid:18) rs (cid:19) − r − bℓ + 1 (cid:19) = Ω (cid:16) n rr − ( − ℓ ) (cid:17) . Proof.
Let G = ( V, E ) be a complete graph K n , let C be a set of colors, and let χ : E → C be an ( rs, (cid:0) rs (cid:1) − r − bℓ + 1)-coloring of G , where s = | V ( K ℓ ,b ) | = 3 + b + 3( ℓ − b . Since χ is an ( rs, (cid:0) rs (cid:1) − rs + 3)-coloring, by Proposition 2.2, there exists a pruned r -th energy graph ~G = ( ~V , ~E ) of G .Assume to the contrary that there is a copy ~K of K ℓ , rbℓ in ~G . Let { ~a , ~a , ~a } be the setof vertices of degree 30 rbℓ in ~K , { ~b , . . . ,~b rbℓ } be the set of vertices of degree 3 in ~K , and ~x j,i and ~y j,i denote the first and last vertices (that are not ~a j or ~b i ) along the ~a j – ~b i geodesicin ~K . We will let ~P j,i denote this geodesic. Let ~Q i denote the subgraph of ~K induced by S j =1 V ( ~P j,i ), which we call a page .We may select a sequence of distinct pages ~Q i , . . . , ~Q i b (1+ ℓ ) such that ~x j,i p has no coordinatesin common with any vertex of ~Q i p ′ for any 1 ≤ p ′ < p ≤ b (1 + ℓ ) and 1 ≤ j ≤
3. This ispossible because there are at most p − X p ′ =1 | V ( ~Q i p ′ ) | ≤ (3 ℓ + 1)( p − ≤ (3 ℓ + 1) · b (1 + ℓ )20ertices whose coordinates we must exclude from choices of ~x j,i p . Since the vertices ~x j,i withthe same j are all distance two from each other, they must have distinct coordinates, so atmost (3 r )(3 ℓ + 1)(3 b (1 + ℓ )) < rbℓ choices for ~Q i p are excluded, and we can choose thissequence i , . . . , i b (1+ ℓ ) . We reorder the pages so that these pages come first, that is, i p = p for all 1 ≤ p ≤ b (1 + ℓ ), and we discard the remaining pages.As in the proof of Theorem 4.1, our goal is to apply Theorem 3.2. It would be convenientto apply Theorem 3.2 to a graph ~T ⊆ ~G where ~T ∼ = K ℓ ,b , however the total savings S + m X i =1 ( d i > (cid:18) r − d i r − (cid:19) from (9) may not be large enough for our purposes in this case. Instead, we will choose agraph ~T more carefully. In general, we will look at the pages ~Q i one at a time, and if addinga page to ~T will increase the total savings by 2 r (the amount we would expect from revealingthis page if we were revealing color-isomorphic copies of K ℓ ,b in all r coordinates), we willdo so. If this does not happen, we will instead try to find a path ~P j,i , such that adding thispath increases (9) by r .In order to help keep track of the total savings, we will build the graph ~T in b ‘chapters,’where each chapter is three consecutive pages of ~K . Depending on how a chapter interactsin G with the previously-revealed chapters, we determine which parts of those three pagesto add to ~T . Thus, we will use the pages ~Q i for 1 ≤ i ≤ b in our b chapters. The remaining3 bℓ pages will yield a π ( S bi =1 ~Q i )-reservoir of size 3 bℓ . Indeed, note that by the ordering weplaced on the pages, ~R := { ~x ,i | b + 1 ≤ i ≤ bℓ } is a π ( S bi =1 ~Q i )-reservoir with source ~a .Let ~T be the empty graph on vertex set V ( ~T ) = { ~a , ~a , ~a } , and let H (0) := π ( ~T ). Wecall an ordering on any subset of E ( ~T ) consistent if it satisfies the following properties: • the edges on any path ~P j,i are ordered canonically from ~a j to ~b i , • the edges of a path ~P j,i appear before the edges of ~P j ′ ,i ′ whenever i < i ′ or i = i ′ and j < j ′ .Note that the first property implies that, for subsets of E ( ~T ) which are unions of paths,consistent orderings are H (0) -compatible.We will assume throughout the rest of the proof that all subgraphs are revealed accordingto a consistent ordering, and will therefore drop any reference to orderings in our notation.We will recursively define graphs ~T , ~T , . . . , ~T b ⊆ ~K such that the following hold: • sav( ~T i , H (0) ∪ S i − j =1 π ( ~T j )) ≥ r , • ~T i = ~P j ,i ∪ ~P j ,i ∪ ~P j ,i , where 3 i − ≤ i , i , i ≤ i , and j , j , j ∈ [3].21ore specifically, ~T i will either be equal to one of the pages ~Q i − , ~Q i − , ~Q i , or ~T i willcontain the first path from each of these three pages. Let us assume that ~T i ′ has been chosento satisfy the above requirements for all i ′ < i , and let us define for each i ′ ≤ i , H ( i ′ ) := H (0) ∪ i ′ − [ j =1 π ( ~T j ) . If for some i ′ with 3 i − ≤ i ′ ≤ i , we havesav (cid:16) ~Q i ′ , H ( i − (cid:17) ≥ r, (13)then set ~T i := ~Q i ′ . (If there is more than one choice of i ′ , choose one arbitrarily.) Otherwise,set ~T i := S ii ′ =3 i − P ,i ′ .Note that when ℓ = 2, we have sav( ~Q i ′ , H ( i − ) ≥ r for all 3 i − ≤ i ′ ≤ i . To see why,note that in this case, ~x j,i ′ = ~y j,i ′ for each 1 ≤ j ≤
3, so not only do we have ~x j,i ′ / ∈ H ( i − ,but also ~x j,i ′ = ~x j ′ ,i ′ for j = j ′ . Therefore, π k ( ~a ~x ,i ′ ) is not in H ( i − ∪ π ( ~P ,i ′ ) for any k , andLemma 3.4 guarantees that S ( ~P ,i , H ( i − ) ≥ r . Repeating this argument with ~P ,i ′ showsthat S ( ~Q i ′ , H ( i − ) ≥ r .When ℓ ≥
3, the following technical lemma ensures that our choice of ~T i gives at least 2 r total savings. Lemma 4.3.
Let i − ≤ i ′ ≤ i . If sav (cid:16) ~Q i ′ , H ( i − (cid:17) < r, then, sav ~P ,i ′ , H ( i − ∪ i ′ − [ j ′ =3 i − π ( ~P ,j ′ ) ! ≥ r/ . We delay the proof of Lemma 4.3 until we finish this proof of Theorem 4.2. Lemma 4.3implies that for all 1 ≤ i ≤ b , there is a choice of ~T i such thatsav( ~T i , H ( i − ) ≥ r. Let ~T := S bi =1 ~T i . Then, by repeated application of (10), we have thatsav( ~T , H (0) ) = b X i =1 sav( ~T i , H ( i − ) ≥ rb. Furthermore, recall that ~R is a π ( S bi =1 ~Q i )-reservoir, and since H (0) ∪ π ( ~T ) ⊆ π ( S bi =1 ~Q i ), ~R is an H (0) ∪ π ( ~T )-reservoir of size 3 bℓ ≥ ⌈ D ( ~T , H (0) ) /r ⌉ . Since | E ( ~T i ) | = 3 ℓ , we have22 E ( ~T ) | = 3 bℓ . Then Theorem 3.2 (applied with H (0) as H and 2 rb as t ), gives us a graph H ∗ ⊆ G with | V ( H ∗ ) | ≤ | V ( H (0) ) | + r | E ( ~T ) | − t ≤ r + 3 rbℓ − rb = r | V ( K ℓ ,b ) | and at least ( r − | E ( ~T ) | = ( r − bℓ repetitions, contradicting the choice of coloring on G . Thus, ~G is K ℓ , rbℓ -free, so | E ( ~G ) | ≤ ex (cid:0) n r , K ℓ , rbℓ (cid:1) = O (cid:16) n r + r ℓ (cid:17) . Therefore, by Propositions 2.1 and 2.2, | C | = Ω (cid:16) n rr − ( − ℓ ) (cid:17) . (cid:4) Proof of Lemma 4.3.
Fix i ′ with 3 i − ≤ i ′ ≤ i , and assume thatsav (cid:16) ~Q i ′ , H ( i − (cid:17) < r. Our goal is to show that the path ~P ,i ′ gives at least 2 r/ Lemma 4.4. If N k ( ~P ,i ′ , H ( i − ) = ℓ , then S k ( ~P ,i ′ ∪ ~P ,i ′ , H ( i − ∪ π ( ~P ,i ′ )) ≥ .Proof. To simplify notation, let H = H ( i − , ~b = ~b i ′ , ~P j = ~P j,i ′ , ~x j = ~x j,i ′ , and ~y j = ~y j,i ′ for j ∈ [3]. By Observation 3.3, we are done unless S k ( ~P , H ∪ π ( ~P )) ≤
1. Since π k ( ~P ) is a pathending outside of H and π k ( ~y ) = π k ( ~y ), the edge π k ( ~y ~b ) does not appear in H ∪ π k ( ~P ).Thus by Lemma 3.4, S k ( ~P , H ∪ π ( ~P )) = 1.Let ~e be the edge of ~P at which we get this savings, and let ~P ′ be the path containing all ofthe edges of ~P coming before and including ~e in the fixed consistent ordering. Then π ( ~P ′ )must contain π k ( ~y ~b ), or else by Lemma 3.4, ~P would get an additional savings after ~e . Since π k ( ~x ) H and by the P -preserving property, the walk π k ( ~P ′ ) follows the path π k ( ~P ) forsome (possibly zero) number of edges, then leaves H ∪ π ( ~P ), and afterwards first encountersanother vertex of H ∪ π ( ~P ) or a previous vertex of ~P with edge π k ( ~e ). The first edge of π k ( ~P ) that is not in H ∪ π ( ~P ) cannot be from π k ( ~b ) to π k ( ~y ) or π k ( ~y ), since otherwise ~P ′ would have length greater than ℓ . Thus ~e = ~y ~b . Furthermore, after ~e , the edges of ~P mustbe contained in H ∪ π ( ~P ∪ ~P ′ ). Since ~P has length at most ℓ , we have ~P ′ = ~P , and itcannot be that π k ( ~y ) is on H ∪ π ( ~P ). Thus, π k ( ~y ~b ) is not in H ∪ π ( ~P ∪ ~P ), since otherwise π k ( ~e ) = π k ( ~y ~b ) = π k ( ~y ~b ), but π k ( ~y ) = π k ( ~y ). By Lemma 3.4, S k ( ~P , H ∪ π ( ~P ∪ ~P )) ≥ (cid:4) Lemma 4.5. If S k ( ~P ,i ′ , H ( i − ) = 1 and D k ( ~P ,i ′ , H ( i − ) ≤ , then S k ( ~Q i ′ , H ( i − ) ≥ . roof. Let H = H ( i − , ~P j = ~P j,i ′ , ~b i ′ = ~b , ~x j = ~x j,i ′ , and ~y j = ~y j,i ′ for j ∈ [3]. We aredone unless S k ( ~P ∪ ~P , H ∪ π ( ~P )) = 0, and since ~P and ~P terminate in H ∪ π ( ~P ), thisimplies that when revealing ~P and ~P , we only have delayed vertices, by Lemma 3.4. Since ~x , ~x H , by the P -preserving property, π k ( ~P ) and π k ( ~P ) trace π k ( ~P ) until the pointwhere ~P gets its vertex savings.If π k ( ~b ) ∈ H , then since D k ( ~P , H ) ≤ π k ( ~P ) is disjoint from H until the last two steps of ~P . But since π k ( ~P ) traces π k ( ~P ) until the last two steps, this implies π k ( ~y ) = π k ( ~y ), acontradiction. Thus, we can assume π k ( ~b ) H . Now, if the degree of π k ( ~b ) in H ∪ π ( ~P ) isless than 3, either π k ( ~y ~b ) or π k ( ~y ~b ) was not revealed when we revealed ~P , so via Lemma 3.4,there will be a second vertex savings when either ~P or ~P are revealed.The only remaining case to consider is when π k ( ~b ) is not in H and has degree at least 3 in H ∪ π ( ~P ). This implies that π k ( ~P ) never returns to a vertex in H and visits π k ( ~b ) twice.There are only two walks of length ℓ tracing π k ( ~P ) ending in π k ( ~b ), one of which is π k ( ~P )itself. Since the π k ( ~y j ) are distinct, this means that one of π k ( ~P ) and π k ( ~P ) does not trace π k ( ~P ), and thus gives another savings. (cid:4) With these two lemmas in hand, let us return to the proof of Lemma 4.3.Let H ∗ := H ( i − ∪ S i ′ − j ′ =3 i − π ( ~P ,j ′ ). We wish to calculate the total savings for the page ~Q i ′ ,so we will consider the r different coordinates of this page separately.We will call the coordinate k ∈ [ r ] good if S k ( ~Q i ′ , H ( i − ) ≥
2, and bad otherwise. Notethat if all k coordinates are good, sav( ~Q i ′ , H ( i − ) ≥ r , so we are done unless we haveat least one bad coordinate. Fix k ∈ [ r ] and assume coordinate k is bad. By the con-trapositive of Lemma 4.4, N k ( ~P ,i ′ , H ( i − ) < ℓ . Recall that π k ( ~x ,i ′ ) H ( i − , so we have N k ( ~P ,i ′ , H ( i − ) >
0. Since new vertices can only be followed by new vertices or savings,then S k ( ~P ,i ′ , H ( i − ) = 1. Furthermore, by Lemma 4.5, D k ( ~P ,i ′ , H ( i − ) ≥
2, so the vertexsavings does not happen on the last or second-to-last step of revealing ~P ,i ′ .We claim that after this vertex savings happens, every other step of revealing ~Q i ′ will consistof a delayed vertex in coordinate k . Indeed, since there cannot be a second vertex savings incoordinate k , if there is a step that is not a delayed vertex, it must be a new vertex. Firstconsider the possibility that the step in which we reveal the edge ~y ,i ′ ~b i ′ is a new vertex. Inthis case, the edge π k ( ~y ,i ′ ~b i ′ ) cannot be in H ( i − ∪ π k ( ~P ,i ′ ). Thus, by Lemma 3.4, as wereveal ~P ,i ′ , we encounter a new edge, while π k ( ~b i ′ ) already has been revealed, so we get asecond savings in coordinate k , contradicting that coordinate k is bad.Thus, the final step of revealing ~P ,i ′ is a delayed vertex, and so every step between the vertexsavings and this final step is also a delayed vertex. Furthermore, once ~P ,i ′ is revealed, if E ( π k ( ~P ,i ′ ∪ ~P ,i ′ )) \ E ( H ( i − ∪ π ( ~P ,i ′ )) = ∅ , Lemma 3.4 gives us a second savings in coordinate k , since π k ( ~b i ′ ) has been revealed. Thus, every step after the vertex savings in coordinate k gives us a delayed vertex in coordinate k . 24ow we can calculate sav( ~Q i ′ , H ( i − ). Let there be b ∗ bad coordinates and r ∗ coordinates k such that N k ( ~P ,i ′ , H ( i − ) = ℓ . Note that we can assume b ∗ ≥
1. We claim that we aredone unless r ∗ ≥ ⌊ r/ ⌋ + 1. Indeed, if not, note that since N k is monotone with respect tosubgraphs (see (11)), there are at most ⌊ r/ ⌋ coordinates k such that N k ( ~P , i − i ′ , H ∗ ) = ℓ ,and thus, there are at least r − ⌊ r/ ⌋ ≥ r/ k with S k ( ~P ,i ′ , H ∗ ) ≥
1. Therefore,we have S ( ~P ,i ′ , H ∗ ) ≥ r/ , as desired.Now, since each bad coordinate has a vertex savings, and all other coordinates have at leasttwo vertex savings, S ( ~Q i ′ , H ( i − ) ≥ b ∗ + 2( r − b ∗ ) = 2 r − b ∗ . Furthermore, we established that the last two steps of revealing ~P ,i ′ in each bad coordinategive us a delayed vertex, while in the r ∗ coordinates with no vertex savings in the first path,these last two steps are not delayed vertices, so these two steps contribute at least2 r ∗ r − X i ∗ ( d i ∗ > (cid:18) r − d i ∗ r − (cid:19) , where the sum is taken over all steps i ∗ that correspond to revealing ~P ,i ′ and ~P ,i ′ . ByLemma 4.4, when revealing ~P ,i ′ and ~P ,i ′ , each of the r ∗ coordinates with no savings in thefirst path give us two vertex savings, which occurs after ~P ,i ′ has already been revealed, andconsequently these two steps do not constitute delayed vertices. Thus, X i ∗ ( d i ∗ > (cid:18) r − d i ∗ r − (cid:19) = P i ∗ r − d i ∗ r − ≥ r ∗ r − , where the first equality follows since we have at least one bad coordinate, and in thatcoordinate, there is a delayed vertex in every step i ∗ considered in the sum. We now calculatethe total savings. We have thatsav( ~Q i ′ , H ( i − ) ≥ r − b ∗ + 4 r ∗ r − . (14)Using the fact that b ∗ ≤ r − r ∗ and that r ∗ ≥ ⌊ r/ ⌋ + 1, the bound in (14) is always at least2 r when r ∈ { , } , so we obtain a contradiction in these cases.Now, let us assume that r ∈ { , } . Note that our earlier analysis gives us that b ∗ ≤ b ∗ ≤
2. To see this, let us assume to the contrary that25 ∗ = 3. Now, let us consider sav( ~P ,i ′ , H ∗ ). Due to the monotonicity of delayed vertices withrespect to subgraphs, see (12), we know that when revealing ~P ,i ′ with respect to H ∗ , thefinal two steps still give us delayed vertices in each of the three bad coordinates. This impliesthat S ( ~P ,i ′ , H ∗ ) ≥ , since the first step of revealing ~P ,i ′ must constitute a new vertex in every coordinate, and sowe must have at least one vertex savings in each bad coordinate before we can encounter thedelayed vertices in the last two steps of revealing the path (Lemma 3.4). Furthermore, thetotal vertex savings must be exactly 3 here, since 2 r/ ≤ r ∈ { , } , and we are doneunless ~P ,i ′ does not have 2 r/ r ∗ coordinates that are not bad give us new vertices at every step of ~P ,i ′ , thus, basedon the delayed vertices in the bad coordinates, we can see thatsav( ~P ,i ′ , H ∗ ) ≥ r − ≥ ≥ r/ . Otherwise, b ∗ ≤ r ∗ ≥ ⌊ r/ ⌋ + 1, so we see that in thiscase, the right side of (14) is larger than 2 r for r ∈ { , } , completing the proof. (cid:4) We believe that Theorem 4.2 can be generalized further, although more case analysis andnew ideas are needed.
Conjecture 5.1.
For positive integers a, b, ℓ, r with a ≤ b and rr − (1 − a − aℓ ) ≥ , the graph K ℓa,b is r -nice. If a graph F is r -nice, then upper bounds for ex( n, F ) yield lower bounds on f ( n, p, q ).As mentioned in Section 1, in the contrapositive, this means upper bounds on f ( n, p, q )yield lower bounds on ex( n, F ). It could be interesting to further explore this connection.For instance, Theorem 4.1 shows that C k is r -nice, so obtaining new upper bounds on f ( n, rk, (cid:0) rk (cid:1) − r − k + 1) would give lower bounds on ex( n, C k ), a well-known hardproblem. The smallest interesting example of this implication is for r = 2 and k = 4. Proving f ( n, , (cid:0) (cid:1) −
7) = O ( n / ) would imply that ex( n, C ) = Θ( n / ). Currently the best upperbound for f ( n, , (cid:0) (cid:1) −
7) is O ( n / ), given by the local lemma bound in (1).Theorem 1.1 from [15] exploited another connection between f ( n, p, q ) and Tur´an numbers(not using the color energy graph). This connection appeared implicitly in [15], so we makethe connection explicit here. Lemma 5.2.
Let < γ < be fixed, and let F be a bipartite graph with bipartition V ( F ) = A ∪ B which contains at least two edges. Suppose that every subgraph of K n,n γ with (cid:0) n (cid:1) / ( | A | +26 E ( F ) | ) edges contains a copy of F with A on the side of size n and B on the side of size n γ . Then, f (cid:18) n, | A | + | E ( F ) | , (cid:18) | A | + | E ( F ) | (cid:19) − ( | E ( F ) | − | B | ) + 1 (cid:19) ≥ n γ . Proof.
Let G = ( V, E ) be a complete graph K n , let C be a set of colors, and let χ : E → C be an ( | A | + | E ( F ) | , (cid:0) | A | + | E ( F ) | (cid:1) − ( | E ( F ) | − | B | ) + 1)-coloring of G . Note that when p := | A | + | E ( F ) | , this coloring is above the linear threshold, (cid:0) p (cid:1) − p + 3, so the color degreesin G are bounded by p − | A | + | E ( F ) | −
1. Suppose for the sake of contradiction that | C | ≤ n γ . First, form G ′ by deleting from G all edges whose color appears on fewer than n − γ edges. In doing so, we delete at most | C | n − γ ≤ n γ = o ( n ) edges total, so G ′ stillhas Ω( n ) edges.We form the color incidence graph , which is a bipartite graph with parts V and C , and anedge between v ∈ V and c ∈ C if there is an edge of G ′ incident to v with color c . Sinceeach vertex of G ′ is incident to at most | A | + | E ( F ) | − (cid:0) n (cid:1) / ( | A | + | E ( F ) | ) edges in the color incidence graph. By our hypothesis, this impliesthat there is a copy of F in the color incidence graph with A ⊆ V and B ⊆ C .This copy of F gives us a clique with many color repetitions in G ′ . In G ′ , for each vertex v ∈ A , there is a star, S v , centered at v with d F ( v ) edges, each of which are colored witha color from B . Let S := S v ∈ A S v . If each of the stars S v are pairwise edge-disjoint for all v ∈ A , then S contains at most X v ∈ A | V ( S v ) | = | A | + X v ∈ A d F ( v ) = | A | + | E ( F ) | vertices, and exactly X v ∈ A | E ( S v ) | − | B | = | E ( F ) | − | B | color repetitions, which contradicts our choice of coloring. Therefore, these stars are notpairwise edge-disjoint, and we let a = P v ∈ A | E ( S v ) | − | E ( S ) | . Then S has at most X v ∈ A | V ( S v ) | − a = | A | + X v ∈ A d F ( v ) − a = | A | + | E ( F ) | − a vertices and exactly | E ( S ) | − | B | = | E ( F ) | − a − | B | color repetitions. Since every color class in G ′ has at least n − γ > (cid:0) | V ( S ) | (cid:1) + a edges, we canchoose a set of a edges, disjoint from E ( S ) that are colored with a color in B . These a edgesspan at most 2 a vertices, so adding them to S gives us a graph on at most | A | + | E ( F ) | vertices with | E ( F ) | − | B | color repetitions, again a contradiction. Thus, | C | > n γ . (cid:4) F = K s,t [13]. In this case, Lemma 5.2 recovers Theorem 1.1. While other upper boundsfor asymmetric bipartite Tur´an numbers are known (for example, for theta graphs), we havefound no further applications of Lemma 5.2 which give improvements on existing bounds.It could be fruitful to further explore this connection between f ( n, p, q ) and asymmetricbipartite Tur´an numbers. References [1] B. Bukh and M. Tait. Tur´an numbers of theta graphs.
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