aa r X i v : . [ m a t h . C O ] O c t Constrained Ramsey Numbers
Po-Shen Loh ∗ Benny Sudakov † Abstract
For two graphs S and T , the constrained Ramsey number f ( S, T ) is the minimum n suchthat every edge coloring of the complete graph on n vertices (with any number of colors) hasa monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T . Here,a subgraph is said to be rainbow if all of its edges have different colors. It is an immediateconsequence of the Erd˝os-Rado Canonical Ramsey Theorem that f ( S, T ) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f ( S, T )for various types of parameters. When S and T are both trees having s and t edges respectively,Jamison, Jiang, and Ling showed that f ( S, T ) ≤ O ( st ) and conjectured that it is always at most O ( st ). They also mentioned that one of the most interesting open special cases is when T is apath. In this paper, we study this case and show that f ( S, P t ) = O ( st log t ), which differs onlyby a logarithmic factor from the conjecture. This substantially improves the previous bounds formost values of s and t . The Erd˝os-Rado Canonical Ramsey Theorem [6] guarantees that for any m , there is some n such thatany edge coloring of the complete graph on the vertex set { , . . . , n } , with arbitrarily many colors,has a complete subgraph of size m whose coloring is one of the following three types: monochromatic,rainbow, or lexical. Here, a subgraph is rainbow if all edges receive distinct colors, and it is lexicalwhen there is a total order of its vertices such that two edges have the same color if and only if theyshare the same larger endpoint.Since the the first two types of colorings are somewhat more natural, it is interesting to studythe cases when we can guarantee the existence of either monochromatic or rainbow subgraphs. Thismotivates the notion of constrained Ramsey number f ( S, T ), which is defined to be the minimum n such that every edge coloring of the complete graph on n vertices (with any number of colors) ∗ Department of Mathematics, Princeton University, Princeton, NJ 08544. E-mail: [email protected] .Research supported in part by a Fannie and John Hertz Foundation Fellowship, an NSF Graduate Research Fellowship,and a Princeton Centennial Fellowship. † Department of Mathematics, UCLA, Los Angeles, CA 90095. Email: [email protected] . Research sup-ported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by anAlfred P. Sloan fellowship. S or a rainbow subgraph isomorphic to T . It is animmediate consequence of the Canonical Ramsey Theorem that this number exists if and only if S is a star or T is acyclic, because stars are the only graphs that admit a simultaneously lexical andmonochromatic coloring, and forests are the only graphs that admit a simultaneously lexical andrainbow coloring.The constrained Ramsey number has been studied by many researchers [1, 3, 4, 7, 8, 10, 13, 14, 18],and the bipartite case in [2]. In the special case when H = K ,k +1 is a star with k + 1 edges, coloringswith no rainbow H have the property that every vertex is incident to edges of at most k differentcolors, and such colorings are called k -local. Hence f ( S, K ,k +1 ) corresponds precisely to the local k -Ramsey numbers, r k loc ( S ), which were introduced and studied by Gy´arf´as, Lehel, Schelp, and Tuzain [11]. These numbers were shown to be within a constant factor (depending only on k ) of theclassical k -colored Ramsey numbers r ( S ; k ), by Truszczy´nski and Tuza [16].When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling [13]conjectured that f ( S, T ) = O ( st ), and provided a construction which showed that the conjecture, iftrue, is best possible up to a multiplicative constant. Here is a variant of such construction, which wepresent for the sake of completeness, which shows that in general the upper bound on f ( S, T ) cannotbe brought below (1 + o (1)) st . For a prime power t let F t be the finite field with t elements. Considerthe complete graph with vertex set equal to the affine plane F t × F t , and color each edge based on theslope of the line between the corresponding vertices in the affine plane. The number of different slopes(hence colors) is t + 1, so there is no rainbow graph with t + 2 edges. Also, monochromatic connectedcomponents are cliques of order t , corresponding to affine lines. Therefore if Ω(log t ) < s < t , wecan take a random subset of the construction (taking each vertex independently with probability s/t ) to obtain a coloring of the complete graph of order (1 + o (1)) st with t + 1 colors in which allmonochromatic connected components have size at most (1 + o (1)) s .Although Jamison, Jiang, and Ling were unable to prove their conjecture, they showed that f ( S, T ) = O ( st · d T ) ≤ O ( st ), where d T is the diameter of T . Since this bound clearly gets weakeras the diameter of T grows, they asked whether a pair of paths maximizes f ( S, T ), over all treeswith s and t edges, respectively. This generated much interest in the special case when T is a path P t . In [18], Wagner proved that f ( S, P t ) ≤ O ( s t ). This bound grows linearly in t when s is fixedbut still has order of magnitude t for trees of similar size. Although Gy´arf´as, Lehel, and Schelp [10]recently showed that for small t (less than 6), paths are not the extremal example, they remain oneof the most interesting cases of the constrained Ramsey problem.In this paper we prove the following theorem which agrees with the conjecture, up to a logarithmicfactor and the fact that T is a path. It significantly improves the previous bounds for most values of s and t , and in particular gives the first sub-cubic bound for the case when the monochromatic treeand rainbow path are of comparable size. Theorem 1.1.
Let S be any tree with s edges, and let t be a positive integer. Then, for any n ≥ st log t , every coloring of the edges of the complete graph K n (with any number of colors) ontains a monochromatic copy of S or a rainbow t -edge path. This supports the conjectured upper bound of O ( st ) for the constrained Ramsey number of apair of trees. With Oleg Pikhurko, the second author obtained another result which provides furtherevidence for the conjecture. This result studies a natural relaxation of the above problem, in whichone wants to find either a monochromatic copy of a tree S or a properly colored copy of a tree T . Itappears that in this case the logarithmic factor can be removed, giving an O ( st ) upper bound. Weview this result as complementary to our main theorem, and therefore have included its short proofin the appendix to our paper.We close this section by comparing our approach to Wagner’s, as the two proofs share somesimilarities. This will also lead us to introduce one of the the main tools that we will use later.Both proofs find a structured subgraph G ′ ⊂ G in which one may direct some edges in such a waythat directed paths correspond to rainbow paths. Wagner’s approach imposes more structure on G ′ , which simplifies the task of finding directed paths, but this comes at the cost of substantiallyreducing | G ′ | . In particular, his | G ′ | is s times smaller than | G | , which contributes a factor of s tohis ultimate bound O ( s t ). We instead construct a subgraph with weaker properties, but of orderwhich is a constant fraction of | G | (hence saving a factor of s in the bound). This complicates theproblem of finding the appropriate directed paths, which we overcome by using the following notionof median order : Definition.
Let G be a graph, some of whose edges are directed. Given a linear ordering σ =( v , . . . , v n ) of the vertex set, a directed edge −−→ v i v j is said to be forward if i < j , and backward if i > j . If σ maximizes the number of forward edges, it is called a median order . Median orders were originally studied for their own sake; for example, finding a median order fora general digraph is known to be NP-hard. More recently, Havet and Thomass´e [12] discovered thatthey are a powerful tool for inductively building directed paths in tournaments (complete graphswith all edges directed). Their paper used this method to produce a short proof of Dean’s conjecture(see [5]) that every tournament has a vertex whose second neighborhood is at least as large as thefirst. Havet and Thomass´e also used a median order to attack Sumner’s conjecture (see [19]) thatevery tournament of order 2 n − n . They succeeded in provingthis conjecture precisely for arborescences (oriented trees where every vertex except the root hasindegree one) and within a factor-2 approximation for general oriented trees.The only property that they used is the so-called feedback property : if σ = ( v , . . . , v n ) is a medianorder, then for any pair i < k , the number of forward edges −−→ v i v j with i < j ≤ k is at least the numberof backward edges ←−− v i v j with i < j ≤ k . This property is easily seen to be true by comparing σ to thelinear order σ ′ = ( v , v , . . . , v i − , v i +1 , v i +2 , . . . , v k , v i , v k +1 , v k +2 , . . . , v n ), which was obtained from σ by moving v i to the position between v k and v k +1 . As an illustration of the simple power of thisproperty, consider the following well-known result, which we will in fact use later in our proof.3 laim. Every tournament has a directed Hamiltonian path.
Proof.
Let σ = ( v , . . . , v n ) be a median order. For each i , the edge v i v i +1 is directed in some waybecause we have a tournament, and so the feedback property applied with k = i + 1 implies that itis in fact a forward edge −−−→ v i v i +1 . Therefore, ( v , . . . , v n ) is already a directed path, so we are done. (cid:3) Let us assume for the sake of contradiction that n ≥ st log t , but there is no monochromaticcopy of S and no rainbow t -edge path. In the past papers on the constrained Ramsey numbers oftrees [13, 18], and in this work, the following well-known crude lemma is the only method used toexploit the nonexistence of a monochromatic S . Its proof follows from the observation that everygraph with average degree ≥ s has an induced subgraph with minimum degree ≥ s . Lemma 2.1.
Let S be a tree with s edges, and let G = ( V, E ) be a simple graph, edge-colored with k colors, with no monochromatic subgraph isomorphic to S . Then | E | < ks | V | . The rest of the proof of our main theorem roughly separates into two main steps. First, we finda structured subgraph G ′ ⊂ G whose order is within a constant factor of | G | . We aim to arrive ata contradiction by using G ′ to construct a rainbow t -edge path. The structure of G ′ allows us todirect many of its edges in such a way that certain directed paths are automatically rainbow. Inthe second step, we use the median order’s feedback property to find many directed paths, which wethen connect into a single long rainbow path using the structure of G ′ . In this section, we show how to find a nicely structured subset of our original graph, at a cost of aconstant factor reduction of the size of our vertex set. We then show how the search for a rainbowpath reduces to a search for a particular collection of directed paths.
Lemma 2.2.
Let S be a tree with s edges and t be a positive integer. Let G be a complete graph on n ≥ st vertices whose edges are colored (in any number of colors) in such a way that G has nomonochromatic copy of S and no rainbow t -edge path.Then there exists a set R of “rogue colors”, a subset U ⊂ V ( G ) with a partition U = U ∪ · · · ∪ U r ,an association of a distinct color c i R to each U i , and an orientation of some of the edges of theinduced subgraph G [ U ] , which satisfy the following properties: (i) | U | > n , | R | < t , and each | U i | < s . (ii) For any edge between vertices x ∈ U i and y ∈ U j with i = j , if it is directed −→ xy , its color is c i ,if it is directed −→ yx , its color is c j , and if it is undirected, its color is in R . iii) For any pair of vertices x ∈ U i and y ∈ U j (where i may equal j ), there exist at least t vertices z U such that the color of the edge xz is c i and the color of yz is c j . Proof.
Let us say that a vertex v is t -robust if for every set F of t colors, there are at least n/ v that are not in any of the colors in F . Let V ⊂ V be the set of t -robust vertices.We will need a lower bound on | V | , but this is just a special case of Lemma A.2 (whose shortproof appears in the appendix). Substituting the values a = n/ b = t into this lemma gives | V ( G ) \ V | ≤ ts + n/ < n/ | V | ≥ n/ P be a rainbow path of maximal length in G such that at least one of its endpoints isin V , and let R be the set of colors of the edges of P . | R | < t by the assumption that G contains norainbow t -edge path. Let B be the set of vertices that have at least n/
15 adjacent edges in a colorin R . Then G contains at least | B | n/
30 edges with colors in R . On the other hand, by applyingLemma 2.1 to the subgraph of G determined by taking only the edges with colors in R , we see thatthe total number of edges in G with color in R is less than | R | sn < tsn , and so | B | < st .Let v be an endpoint of P which is in V . Define the sets U i as follows. Let { c , . . . , c r } be thenon- R colors that appear on edges adjacent to v . For each such c i , let U i be the set of vertices thatare not in B or P , and are adjacent to v via an edge of color c i . Set U = U ∪ . . . ∪ U r . We claimthat these designations will satisfy the desired properties.Consider arbitrary vertices x ∈ U i and y ∈ U j , where i may equal j . Since n ≥ t , we have | V \ P | ≥ (2 / n + t , so x, y B imply that there are at least t choices for z ∈ V \ P such that bothof the edges xz and yz have colors not in R . Each such xz must be in color c i , or else the extensionof P by the path vxz would contradict maximality of P , and similarly each yz must be in color c j .Finally, U ∩ V = ∅ , because any w ∈ U ∩ V would allow us to extend P by the edge vw . Therefore,we have property (iii).For property (ii), let x ∈ U i and y ∈ U j , with i = j . By property (iii), there exists some vertex z ∈ V \ P such that yz is in color c j . Then the color of the edge xy must be in { c i , c j } ∪ R , or elsethe extension of P by the path vxyz would contradict its maximality. Therefore, we can leave itundirected if the color is in R , and direct it according to property (ii) otherwise.It remains to show property (i). We already established that | R | < t and we can obtain the firstinequality from the construction of V as follows. Since v ∈ V , it is t -robust and so is adjacent to atleast n/ R colors. Therefore, using that n ≥ st we get | U | ≥ n/ − | B | − | P | > n/ − st − t ≥ n/ . For the last part, assume for the sake of contradiction that | U i | ≥ s . Arbitrarily select a subset U ′ i ⊂ U i of size 2 s , and consider the subgraph G ′ formed by the edges of color c i among vertices in U ′ i ∪ V . By the argument that showed property (iii), every edge between U ′ i and V has color in R ∪ { c i } . So, since U i ∩ B = ∅ , every x ∈ U ′ i is adjacent to at least | V | − n/ ≥ (2 / | V | verticesin V via edges of color c i . Therefore, using that | V | / ≥ s = | U ′ i | , we have e ( G ′ ) ≥ | U ′ i | · (2 / | V | = (4 / s | V | = s ( | V | + (1 / | V | ) ≥ s · v ( G ′ ) . G ′ has a copy of tree S , which is monochromatic by construction of G ′ .This contradiction completes the proof of the last part of property (i), and the proof of the lemma. (cid:3) The partially directed subgraph of Lemma 2.2 allows us to find rainbow paths by looking forcertain types of directed paths. For example, if Lemma 2.2 produces U = U ∪ . . . ∪ U m , and wehave found a directed path −−−−−→ v . . . v t with each v i from a distinct U j , then it must be rainbow byproperty (ii) of the construction of U . Unfortunately, the following simple construction of a set withno monochromatic S that satisfies the structure conditions of Lemma 2.2 shows that we cannot hopeto obtain our rainbow path by searching for a single (long) directed path: re-index { U i } with orderedpairs as { U ij } h,t − i =1 ,j =1 , let all | U ij | = s/
3, for all 1 ≤ i < j ≤ h direct all edges between any U i, ∗ and U j, ∗ in the direction U i, ∗ → U j, ∗ , and for all 1 ≤ i ≤ j < t and 1 ≤ k ≤ h color all edges between U k,i and U k,j in color r i , where R = { r , . . . , r t − } . Although it is clear that this construction has nodirected paths longer than h = O (cid:0) | U | st (cid:1) , it is also clear that one could build a long rainbow path bycombining undirected edges and directed paths. The following lemma makes this precise. Lemma 2.3.
Let U = U ∪ . . . ∪ U m be a subset of V ( G ) satisfying the structural conditions of Lemma2.2, and let R be the associated set of rogue colors. Suppose we have a collection of r < t edges { u i v i } ri =1 in G [ U ] whose colors are distinct members of R , and a collection of directed paths { P i } ri =0 ,with P i starting at v i for i ≥ . Then, as long as all of the vertices in { u , . . . , u r } ∪ P ∪ . . . ∪ P r belong to distinct sets U j , there exists a rainbow path in G that contains all of the paths P i and allof the edges u i v i . In short, one can link all of the fragments together into a single rainbow path. Proof.
For each i , let w i be the final vertex in the directed path P i . For a vertex v ∈ U , let c ( v )denote the color associated with the set U i that contains v . Since r < t , by property (iii) of Lemma2.2, for each 0 ≤ i < r , there exists a distinct vertex x i U such that the color of the edge w i x i is c ( w i ) and the color of the edge x i u i +1 is c ( u i +1 ). These vertices x i together with paths P i form apath P of distinct vertices, which we will now prove is rainbow.Note that our linking process only adds edges with non-rogue colors. Since we assumed thatthe u i v i have distinct colors, and the edges of the P i are directed paths (hence with non-roguecolors), it is immediate that P has no duplicate rogue colors. Also note that among all directededges in { P i } , no pair of edges has initial endpoint in the same U j by assumption. Therefore, theyall have distinct colors by property (ii) of Lemma 2.2. Furthermore, none of these directed edgesoriginates from any point in any U j that intersects { u , . . . , u r , w , . . . , w r } , so they share no colorswith C ′ = { c ( u ) , . . . , c ( u r ) , c ( w ) , . . . , c ( w r ) } ; finally the colors in C ′ are themselves distinct becauseof our assumption that all vertices in { u , . . . , u r } ∪ P ∪ . . . ∪ P r come from distinct U j . This provesthat P is a rainbow path. (cid:3) .2 Finding directed paths Now apply Lemma 2.2, and let us focus on U = U ∪ . . . ∪ U m , which is of size at least n/ ≥ st log t . Let us call the edges which have colors in R “rogue edges.” Note that if all edges weredirected (i.e., we have a tournament), then the existence of a long directed path follows from thefact that every tournament has a Hamiltonian path. The main issue is the presence of undirectededges. We treat these by observing that each undirected edge must have one of | R | < t rogue colors.Then, we use the machinery of median orders to repeatedly halve the number of rogue colors, at theexpense of losing only O ( st ) vertices each time. This is roughly the source of the log t factor in ourfinal bound.Now we provide the details to make the above outline rigorous. Applying Lemma 2.1 to thesubgraph consisting of all rogue edges, we see that the average rogue degree (number of adjacentrogue edges) in G [ U ] is at most 2 s | R | ≤ st . So, we can delete all vertices in U with rogue degree atleast 4 st at a cost of reducing | U | by at most half. Let us also delete all edges within each U i for thesake of clarity of presentation. Note that the reduced U still has size at least 180 st log t . Let σ bea median order for this partially directed graph induced by U . We will use the feedback property tofind directed paths (and this is the only property of median orders that we will use).We wish to apply Lemma 2.3, so let us inductively build a matching of distinct rogue colors, andaccumulate a bad set that we call B and which we will maintain and update through the entire proofin this section. Let v be the first vertex according to σ , and start with B = U ℓ , where U ℓ ∋ v .Proceed through the rest of the vertices in the order of σ . For the first stage, stop when we firstencounter a vertex not in B that is adjacent to a rogue edge (possibly several) whose other endpointis also not in B , and call the vertex v . Arbitrarily select one of those rogue edges adjacent to v ,call it e , and call its color r . Since we deleted all edges inside U i , e links two distinct U i and U j .Add all vertices of U i and U j to B . In general, if we already considered all vertices up to v k , continuealong the median order (starting from the vertex immediately after v k ) until we encounter a vertexnot in B that is adjacent to an edge of a new rogue color which is not in { r , . . . , r k } , again withother endpoint also not in B . Call that vertex v k +1 , the edge e k +1 , and its color r k +1 . Add to B all the vertices in the two sets U i which contain the endpoints of e k +1 . Repeat this procedure untilwe have gone through all of the vertices in the order. Suppose that this process produces vertices v , v , . . . , v f . Then, to simplify the statements of our lemmas, also let v f +1 , v f +2 , . . . v f refer tothe final vertex in the median order. Our goal will be to find directed paths from { v i } fi =1 , which viaLemma 2.3, will then extend to a rainbow path.Note that if | B | ≥ st , then the number of vertices in { v } ∪ e ∪ . . . ∪ e f is at least t by property(i). Thus, applying Lemma 2.3 with P i = { v i } , we can produce a rainbow path with at least t edges.Therefore, we may assume for the rest of this proof that | B | < st . Also observe that this argumentimplies that f ≤ t/ { P i } . Lemma 2.4.
Let v be a vertex in U , and let B be a set of size at most st . Then, among the st ertices immediately following v in the median order, there is always some w B such that there isa directed edge from v to w . Proof.
First, note that since we deleted all vertices with rogue degree at least 4 st , more than 4 st ofthe 8 st vertices immediately after v are connected to v by a directed edge. Since we have a medianorder, the feedback property implies that only at most half of those edges can be directed backtowards v ; therefore, there are more than 2 st vertices there that have a directed edge from v . Since | B | < st , at least one of these vertices will serve as our w . (cid:3) Consider the vertices v , v , v , . . . , v ⌊ log2 2 f ⌋ . Since we already established that f ≤ t/
2, this isa list of at most t + 1 vertices, the first and last of which are also the first and last vertices in themedian order. Since U still has at least 180 st log t vertices, the pigeonhole principle implies thatthere must be some pair of vertices { v ℓ , v ℓ } in that list such that the number of vertices betweenthem in the median order is at least 180 st −
2. Thus, the following lemma will provide the desiredcontradiction.
Lemma 2.5.
If there is any ≤ ℓ ≤ f such that there are at least st vertices between v ℓ and v ℓ in the median order, then G has a rainbow t -edge path. Proof.
Suppose we have an ℓ that satisfies the conditions of the lemma. Let S be the first 8 st vertices immediately following v ℓ in the median order, and let S be the next 168 st vertices in themedian order.Let us first build for every i ≤ ℓ a directed path P i from v i to S by repeatedly applying Lemma2.4. Start with each such P i = { v i } , and as long as one of those P i does not reach S , apply thelemma to extend it forward to a new vertex w , and add the set U k containing w to the set of badvertices B . If at any stage we have | B | ≥ st , we can immediately apply Lemma 2.3 to find a rainbowpath with at least t edges, just as in the argument directly preceding the statement of Lemma 2.4.So, suppose that does not happen, and let { w i } ℓ ⊂ S be the endpoints of these paths. We will showthat we can further extend these paths into S by a total amount of at least t , in such a way thatwe never use two vertices from the same set U k . This will complete our proof because Lemma 2.3can link them into a rainbow path with at least t edges.Recall that all of the sets U i had size at most 2 s . Therefore, we can partition S into disjointsets U ′ j with 2 s ≤ | U ′ j | ≤ s , where each U ′ j is obtained by taking a union of some sets U i ∩ S . Wewill design our path extension process such that it uses at most one vertex from each U ′ j , and henceit will also intersect each U k at most once. We use the probabilistic method to accomplish this.Perform the following randomized algorithm, which will build a collection of sets { T i } ℓi =1 . First,activate each U ′ j with probability 1 /
8. Next, for each activated U ′ j , select one of its vertices uniformlyat random, and assign it to one of the T i , again uniformly at random. For each i ≤ ℓ , let T ′ i beobtained from T i by deleting every vertex in B , and every vertex that is not pointed to by a directededge from v i . Finally, let T ′′ i be derived from T ′ i by (arbitrarily) deleting one vertex from every rogue8dge with both endpoints in T ′ i . Observe that now each T ′′ i spans a tournament, so as we saw at theend of the introduction, it contains a directed Hamiltonian path P ′ i . Since w i has a directed edge toevery vertex in T ′′ i , this P ′ i can be used to extend P i . Therefore, if we can construct sets T ′′ i suchthat | T ′′ | + · · · + | T ′′ ℓ | ≥ t , we will be done.Fix an i ≤ ℓ , and let us compute E [ | T ′ i | ]. By the feedback property of a median order, thenumber of (backward) directed edges from S to { w i } is at most half of the number of directed edgesbetween w i and the vertices in S ∪ S which follow it in the median order. Since the latter number isbounded by | S ∪ S | = 176 st , the number of directed edges from S to w i is at most 88 st . Also, thenumber of rogue edges between S and { w i } is at most 4 st because by construction all rogue degreesare bounded by 4 st . Therefore, the number of (forward) directed edges from w i to S is at least168 st − st − st = 76 st . Since we will delete up to 2 st vertices which were from B , the numberof directed edges from w i to vertices in S \ B is at least 74 st . Suppose −−→ w i x is one of these directededges, and suppose that x ∈ U ′ k . The probability that x is selected for T i is precisely · ℓ ·| U ′ k | ≥ · ℓ · s ,and by construction of x , we know that if it is selected for T i , it will also remain in T ′ i . Therefore,by linearity of expectation, E (cid:2) | T ′ i | (cid:3) ≥ st · · ℓ · s = 3716 tℓ To bound E [ | T ′ i | − | T ′′ i | ], observe that the number of rogue colors in the graph spanned by S \ B isless than 2 ℓ , by construction of the sequence { v i } . Therefore, Lemma 2.1 implies that there are lessthan 2 ℓ · s · st rogue edges spanned by S \ B . Consider one of these rogue edges xy . If we selectboth of its endpoints for T i , it will contribute at most 1 (possibly 0) to | T ′ i | − | T ′′ i | ; otherwise it willcontribute 0. Above, we already explained that the probability that the vertex x ∈ U ′ j is selected for T i is precisely · ℓ ·| U ′ k | ≤ · ℓ · s . If x and y come from distinct U ′ j , then the probabilities that they wereboth selected for T i are independent, and otherwise it is impossible that they both were selected.Hence E (cid:2) | T ′ i | − | T ′′ i | (cid:3) ≤ ℓ · s · st · (cid:18) · ℓ · s (cid:19) = 2116 tℓ Therefore, by linearity of expectation, E [ | T ′′ i | ] ≥ t/ℓ , and thus E [ | T ′′ | + · · · + | T ′′ ℓ | ] ≥ t . This impliesthat there exists an instance of our random procedure for which | T ′′ | + · · · + | T ′′ ℓ | ≥ t , so we are done. (cid:3) In our proof, we apply Lemma 2.2 to produce a structured set U = U ∪ . . . ∪ U m of size Ω( st log t ).The argument in Section 2.2 is quite wasteful because, in particular, Lemma 2.5 attempts to build acollection of directed paths with total length ≥ t , but essentially using only the vertices in the medianorder between v ℓ and v ℓ . This dissection of the vertex set into dyadic chunks incurs the logarithmicfactor in our bound. We believe that with a better argument, one might be able to complete theproof using a structured set U = U ∪ . . . ∪ U m of size only Ω( st ). If this were indeed possible, then9emma 2.2 would immediately imply that f ( S, P t ) = O ( st ), because one loses only a constant factorin passing from V ( G ) to U .It would be very interesting to obtain a better bound on f ( S, T ) for general trees T . Our approach,based on the median order, seems particularly promising here since it might be combined with thefollowing result of Havet and Thomass´e [12] on Sumner’s conjecture: every tournament of order 4 n contains every directed tree of order n as a subgraph. A Appendix (by Oleg Pikhurko and Benny Sudakov)
Consider the following variant of the constrained Ramsey number. Let g ( S, T ) be the minimuminteger n such that every coloring of the edges of the complete graph K n contains either a monochro-matic copy of S or a properly colored copy of T . (In contrast, recall that the definition of f ( S, T )requires T to be rainbow). Similarly as for constrained Ramsey numbers, it is easy to see that g ( S, T )exists (i.e., it is finite) if and only if S is a star or T is acyclic. Although there has been little successbounding f ( S, T ) by O ( st ), it turns out that we can prove a quadratic upper bound for g ( S, T ),which is of course no larger than f ( S, T ). Theorem A.1.
Let S and T be two trees with s and t edges, respectively. Then g ( S, T ) ≤ st + t . The following construction shows that the upper bound is tight up to a constant factor. Let S be a path with s + 1 edges and T be a star with t + 1 edges. Then let V , . . . , V t be disjoint sets ofsize ⌊ s/ ⌋ each. Color all edges inside V i and from V i to V j with j > i by color i . This produces agraph on t ⌊ s/ ⌋ vertices with no monochromatic S and no properly colored T .To prove Theorem A.1, we first need the following lemma. Lemma A.2.
Consider an edge coloring of the complete graph which contains no monochromaticcopy of a fixed tree S with s edges. Let U be the set of vertices such that for every u ∈ U , one candelete at most a edges from the graph such that the remaining edges which connect u to the rest ofthe graph have at most b colors. Then | U | ≤ bs + a ) . Proof.
Focus on the subgraph induced by U . Now we can delete at most a edges at every vertexso that the remaining edges at that vertex have at most b colors. Let G be the graph obtainedafter all of these deletions. If m = | U | , then the number of edges of G is at least (cid:0) m (cid:1) − am . Forevery remaining color c , let G c be the subgraph of all edges of color c . By Lemma 2.1, we have e ( G c ) < s · v ( G c ) for each c . Also, since every vertex of G is incident with edges of at most b colors,we have that P c v ( G c ) ≤ bm . Combining all these inequalities we have (cid:18) m (cid:19) − am ≤ X c e ( G c ) < X c s · v ( G c ) ≤ sbm. This implies that m < bs + a ) + 1. (cid:3) roof of Theorem A.1. The proof is by induction on t . The statement is trivial for t = 1 becauseany edge will give us a properly colored T . Now suppose that T is a tree with t > G = K st + t with no monochromatic copy of S . It suffices to show that we canfind a properly colored copy of T . Select an edge ( u, v ) of T such that all neighbors of v except u are leaves v , . . . , v k . Delete v , . . . , v k from T and call the new tree T . The number of edges in T is t = t − k .Let U be the set of vertices of G such that for every u ∈ U one can delete at most t edges from G such that the edges which connect u to the rest of the graph have at most k colors. By the previouslemma | U | ≤ ks + t ), and let W = V ( G ) \ U . Then we have that | W | = 2 st + t − | U | ≥ st + t − t > st + t . Therefore by induction we can find a properly colored copy of the tree T inside W . Let u ′ , v ′ be theimages in this copy of the vertices of u, v of T . By definition of W , the vertex v ′ has edges of atleast k + 1 colors connecting it with vertices outside this copy of T . At least k of these colors aredifferent from that of the edge ( u ′ , v ′ ), so we can extend the tree to a properly colored copy of T . (cid:3) Using a more careful analysis in the above proof, which we omit, one can slightly improve theterm t in Theorem A.1. References [1] N. Alon, T. Jiang, Z. Miller, and D. Pritikin, Properly colored subgraphs and rainbow subgraphsin edge-colorings with local constraints,
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