aa r X i v : . [ m a t h . C O ] J un Ramsey numbers and the size of graphs
Benny Sudakov ∗ Abstract
For two graph H and G , the Ramsey number r ( H, G ) is the smallest positive integer n suchthat every red-blue edge coloring of the complete graph K n on n vertices contains either a redcopy of H or a blue copy of G . Motivated by questions of Erd˝os and Harary, in this note we studyhow the Ramsey number r ( K s , G ) depends on the size of the graph G . For s ≥
3, we prove thatfor every G with m edges, r ( K s , G ) ≥ c (cid:0) m/ log m (cid:1) s +1 s +3 for some positive constant c depending onlyon s . This lower bound improves an earlier result of Erd˝os, Faudree, Rousseau, and Schelp, andis tight up to a polylogarithmic factor when s = 3. We also study the maximum value of r ( K s , G )as a function of m . For two graphs H and G , the Ramsey number r ( H, G ) is the smallest positive integer n such thatany red-blue coloring of the edges of the complete graph K n on n vertices contains either a red copyof G or a blue copy of H . If H = G we usually denote r ( G, G ) by r ( G ). The problem of determiningor accurately estimating Ramsey numbers is one of the central problems in modern Combinatorics,and it has received a considerable attention, see, e.g., [10], [5]. In most cases the Ramsey numberis estimated in terms of the order (number of vertices) of the the graph. However, in the early 80’sErd˝os and Harary asked about the relation between r ( H, G ) and the sizes (number of edges) of thegraphs H and G .The first partial answers to this general problem were obtained by Erd˝os, Faudree, Rousseau, andSchelp [8]. They determined up to a constant factor the minimum value of r ( G ) for all graphs of size m and showed that the order of magnitude of this minimum is Θ( m/ log m ). They also proved thatfor fixed s ≥ c , c such that c m ss +2 < min e ( G )= m r ( K s , G ) < c m s − s . ∗ Department of Mathematics, Princeton University, Princeton, NJ 08544, and Institute for Advanced Study, Prince-ton. E-mail: [email protected]. Research supported in part by NSF CAREER award DMS-0546523, NSFgrant DMS-0355497, USA-Israeli BSF grant, Alfred P. Sloan fellowship, and the State of New Jersey. s = 3). In this case Erd˝os [6] conjectured that the upper bound O ( m / )is closer to the truth. Our first result improves the bounds from [8] and confirms this conjecture. Theorem 1.1
Let s ≥ and let G be a graph with m edges. Then there exists a constant c dependingonly on s such that r ( K s , G ) ≥ c (cid:0) m/ log m (cid:1) s +1 s +3 . On the other hand, there exists a graph G of size m such that r ( K s , G ) ≤ O (cid:16) m s − s / log s − s m (cid:17) . In particular, when s = 3 this determines the minimum value of r ( K , G ) for all graphs G of size m up to a polylogarithmic factor and shows thatΩ m / log / m ! < min e ( G )= m r ( K , G ) < O m / log / m ! . (1)Another specific question which is part of the general problem mentioned in the first paragraph isto bound the maximum value of r ( H, G ) when the graphs H and G have a given size. One of the basicresults in Ramsey Theory is the fact that for the complete graph G with m edges, r ( G ) = 2 Θ( √ m ) .A conjecture of Erd˝os [7] (see also [5]) asserts that there is an absolute constant c such that for anygraph G with m edges, r ( G ) ≤ c √ m . This conjecture is still open. For bipartite graphs it wasproved by Alon, Krivelevich and Sudakov [3]. They also show that for general graphs G with m edges, r ( G ) ≤ c √ m log m for some absolute positive constant c . For the first off-diagonal case Hararyconjectured and Sidorenko [15] proved that r ( K , G ) ≤ m + 1 for any graph G of size m withoutisolated vertices. This inequality is best possible, since r ( K , G ) = 2 m + 1 for any tree with m edges.Thus for s = 3 the graphs which maximize r ( K , G ) are very sparse. However, for s > r ( K s , G ) over all graphs with m edgesone should make G as nearly complete as possible. Motivated by this question we obtain the followinggeneral upper bound on r ( K s , G ) for graphs G of size m . Theorem 1.2
Let s ≥ and let G be a graph with m edges and without isolated vertices. Then thereexists a constant c depending only on s such that r ( K s , G ) ≤ c m s − / log s − m . When G is a clique with m edges it is known by the result of Ajtai, Koml´os and Szemer´edi [1] that r ( K s , G ) ≤ O ( m s − / log s − m ). Hence our estimate has, up to a polylogarithmic factor, similar orderof magnitude to the best known upper bound for off-diagonal Ramsey numbers of cliques.The rest of this short note is organized as follows. In the next section we present proofs of ourmain results. The final section contains some concluding remarks and open problems. Throughoutthe paper we make no attempts to optimize various absolute constants. To simplify the presentation,we often omit floor and ceiling signs whenever these are not crucial. All logarithms are in the naturalbase e . 2 Proofs
To prove Theorem 1.1 we use an approach developed by Krivelevich [13], which is based on proba-bilistic arguments together with large deviation inequalities. The fist inequality we need is a standardbound of Chernoff (see Appendix A in [2]) which states that if X is a binomially distributed randomvariable with parameters m and p , then for every a > P [ X − pm < − a ] ≤ e − a pm . Another large deviation bound, which we use in the proof, was obtained by Erd˝os and Tetali [9] (seealso Chapter 8.4 in [2]).Let Ω be a finite set (in our instance it is the set of edges of a complete graph) and let R be arandom subset of Ω such that P [ ω ∈ R ] = p ω independently for all ω ∈ Ω. Let C i , i ∈ I be subsets ofΩ, where I is some finite index set. For every C i we define A i to be the event that C i ⊆ R . Let X i bethe indicator random variable of event A i and let X = P i ∈ I X i be the number of C i ⊆ R . Finally, let X be the maximum number of pairwise disjoint subsets C i which belong to R . Obviously, X ≤ X .Let µ be the expectation of X ; then Erd˝os and Tetali gave the following bound on the possible sizeof X : P [ X ≥ k ] ≤ µ k k ! ≤ (cid:16) e µk (cid:17) k . Proof of Theorem 1.1.
Let n = s (cid:0) m/ log m (cid:1) s +1 s +3 and consider coloring the edges of the completegraph K n such that each edge is colored randomly and independently red with probability p = s n − s +1 and blue with probability 1 − p . Let G , . . . , G t be all subgraph of K n which are isomorphic to G .The number of such subgraphs t is clearly bounded by the number of injective functions from V ( G )to K n , which in turn is at most the number of permutations on n elements. Thus t ≤ n !. For everysubgraph G i , let X i be the random variable that counts the number of red edges in G i . By definition, X i is binomially distributed with parameters m and p . Hence, by Chernoff’s inequality P [ X i < mp/
2] = P [ X i − mp < − mp/ ≤ e − mp/ . Also for every subgraph G i define Y i to be the number of red cliques of order s which share atleast one edge with G i . Since G i has m edges, the number of s -cliques sharing at least one edge with G i is bounded by mn s − . The probability that an s -clique is red is clearly p ( s ). Therefore E [ Y i ] ≤ mn s − p ( s ) = mpn s − p ( s +1)( s − = (cid:18) s (cid:19) ( s +1)( s − mp ≤ mp s . Let Y ′ i be the maximum number of edge disjoint red s -cliques which share at least one edge with G i Then by the Erd˝os-Tetali inequality we have that P (cid:2) Y ′ i ≥ mp/s (cid:3) ≤ (cid:18) e E [ Y i ] mp/s (cid:19) mp/s ≤ (cid:16) e (cid:17) mp/s ≤ e − mp/s .
3y definition of n and p we have that mp = (3 s n ) s +3 s +1 p log m ≥ s n log n > n log n . Therefore theprobability that for some index i either X i < mp/ Y ′ i ≥ mp/s is bounded by t (cid:0) e − mp/ + e − mp/s (cid:1) ≤ n ! n − n = o (1). In particular there exists a red-blue edge coloring of K n such that for every 1 ≤ i ≤ t ,subgraph G i contains at least mp/ mp/s edge disjoint red s -cliqueseach sharing at least one edge with G i .Fix such a coloring and let Γ be the subgraph of red edges in it. Also let C be the maximum(under inclusion) collection of edge disjoint cliques of order s in Γ. Recolor edges in all cliques from C by blue and denote the remaining red graph by Γ ′ . Note that by recoloring we removed from Γthe maximum collection of edge disjoint s -cliques. Thus Γ ′ contains no clique of order s . On theother hand in every subgraph G i we changed the color of at most (cid:0) s (cid:1) mp/s < mp/ G i had at least mp/ K n isomorphic to G stillhas at least one red edge. This implies that new coloring contains no blue copy of G and no red copyof K s and completes the proof of the first statement of the theorem.To prove the second part, let G be the union of mk ( k − vertex disjoint cliques of order k = m /s (log m ) s − s . By definition, the number of edges in G is at least m . To estimate the Ramseynumber r ( K s , G ) we use the result of Ajtai, Koml´os and Szemer´edi [1] (see also Theorem 12.17 in[4]) which bounds off-diagonal Ramsey numbers. They prove that there exists a constant c such that r ( K s , K k ) ≤ c k s − log s − k . Let n = c k s − log s − k + 2 m/ ( k −
1) = O (cid:16) m s − s / log s − s m (cid:17) and consider any red-blue edge coloring of the complete graph K n . We can assume that there is nored s -clique, or else we are done. Then, since n ≥ r ( K s , K k ), we can find a blue clique of order k .Delete it from the graph and continue this process. Note that as long as we deleted less than mk ( k − cliques of order k the remaining number of vertices is still larger than r ( K s , K k ) and we can find anew blue k -clique. In the end we will find at least mk ( k − blue cliques of size k , i.e., a copy of G . Thisimplies that r ( K s , G ) ≤ O (cid:16) m s − s / log s − s m (cid:17) and completes the proof. (cid:3) Proof of Theorem 1.2.
We prove the theorem by induction on s . Consider the case s = 3.Although one can use results from [8] and [15] to show that r ( K , G ) ≤ O ( m ) we include here thesimple proof that r ( K , G ) ≤ m for the sake of completeness. Clearly we can assume that G isconnected, since r ( K , G ∪ G ) ≤ r ( K , G ) + r ( K , G ). Hence the number of vertices of G is atmost m + 1. Let n = 3 m and suppose that the edges of K n are red-blue colored with no red triangle.Pick the vertex with maximum red degree in this coloring and let X, | X | = t , be the set of its redneighbors. Note that all the edges inside X are blue, since there is no red triangle. Partition thevertices of G into two sets V ( G ) = V ′ ∪ V ′′ , where V ′ consists of the t vertices with the highestdegree. Since the sum of the degrees in G is 2 m , we have that all the vertices in V ′′ have degree atmost 2 m/ ( t + 1). Now we will find the blue copy of G as follows. Embed the vertices of V ′ into X arbitrarily, and then embed the vertices of V ′′ one by one. Given a vertex v ∈ V ′′ , let Y be the set4f vertices of K n where we already embedded neighbors of v . Since the maximum red degree in thecoloring is t and | Y | ≤ d ( v ) ≤ m/ ( t + 1) we have that K n contains at least 3 m − t | Y | ≥ m + 1 − | Y | vertices which are adjacent to all vertices in Y by blue edges. As the total number of vertices of G is at most m + 1, one such vertex is still unoccupied and can be used to embed v . Continuing thisprocess we find a blue copy of G .Now suppose s > r ( K s − , G ) ≤ c m s − / log s − m . Let n = c m s − / log s − m , where c is a sufficiently large constant which depends on c and which we fixlater. Consider a red-blue coloring of the complete graph K n such that there is no red copy of K s . Ifthere is a vertex which has at least d = c m s − / log s − m red neighbors, then this set cannot containa red copy of K s − . Therefore by the induction hypothesis it will contain a blue copy of G and weare done. Thus we can assume that the maximum degree in the red subgraph of K n is at most d .Set k = √ m log m . It is easy to check that, by definition, n = Ω (cid:16) k s − log s − k (cid:17) . Therefore, by choosing c large enough and using the result of Ajtai, Koml´os and Szemer´edi [1] on Ramsey numbers, we getthat n ≥ r ( K s , K k ). Hence there exists a set X of k vertices which spans only blue edges. Againpartition the vertices of G into two sets V ( G ) = V ′ ∪ V ′′ , where V ′ consists of the k vertices withthe highest degree. Since the sum of the degrees in G is 2 m , we have that all the vertices in V ′′ have degree at most 2 m/ ( k + 1). Embed the vertices of V ′ into X arbitrarily, and then embed thevertices of V ′′ one by one as follows. Given a vertex v ∈ V ′′ , let Y be the set of vertices of K n where we already embedded neighbors of v . Since the maximum red degree in the coloring is d and | Y | ≤ d ( v ) ≤ m/ ( k + 1), by choosing sufficiently large c , we have that there are at least n − d | Y | ≥ n − md/ ( k + 1) > c m s − / log s − m − m √ m log m (cid:16) c m s − / log s − m (cid:17) > m vertices in K n which are adjacent to all vertices in Y by blue edges. Note that the total number ofvertices of G is at most 2 m , as it has no isolated vertices. Therefore there exists an unoccupied vertexof K n which is connected to all vertices in Y by blue edges. This vertex can be used to embed v . Inthe end of this procedure we obtain a blue copy of G . This completes the proof of the theorem. (cid:3) • Let H be a graph with v H ≥ e H edges. The density ρ ( H ) of H is defined as ρ ( H ) = e H − v H − . Define also ρ ∗ ( H ) = max H ′ ⊆ H ρ ( H ′ ) . For example, for the complete graph of order s we have ρ ∗ ( K s ) = s +12 . The arguments in theproof of Theorem 1.1 can be used to obtain the following more general result. Since the proofof this statement does not require new ideas and contains somewhat tedious computations weomit it here. 5 heorem 3.1 Let H be a fixed graph. Then there exists a constant c depending only on H such that for every graph G with m edges, r ( H, G ) ≥ c (cid:0) m/ log m (cid:1) ρ ∗ ρ ∗ . In addition to the triangle, this result is nearly tight when H is the complete bipartite graph K p,q with q ≫ p . Indeed it is easy to check from the definition that if p is fixed and q → ∞ then ρ ∗ ( K p,q ) → p . Therefore for every p and ǫ > q such that ρ ∗ ( K p,q )1+ ρ ∗ ( K p,q ) > p p − ǫ .Thus, from Theorem 3.1 we have that r ( K p,q , G ) ≥ Ω (cid:16) m p p − ǫ (cid:17) for every G with m edges. Onthe other hand, from the result of K¨ovari, S´os, Tur´an [12] that K p,q -free graphs on n verticescan have at most O ( n − /p ) edges, it follows that such a graph has an independent set of sizeΩ (cid:0) n /p (cid:1) . This implies that r ( K p,q , K k ) ≤ O ( k p ) (see also [3, 14] for slightly better estimate).Using this bound together with the argument from the proof of the second part of Theorem1.1, we can show that if G is the disjoint union of Θ (cid:0) m p − p +1 (cid:1) cliques of order m / ( p +1) then r ( K p,q , G ) ≤ O (cid:16) m p p (cid:17) . • For s = 3 the lower bound in Theorem 1.1 is tight up to a multiplicative factor of log / m . Itwould be very interesting to close this gap. We think that our upper bound in (1) is closer tothe truth and there exists an absolute constant c such that r ( K , G ) ≥ cm / / log / m for everygraph G of size m . To prove this one might try to use an approach based on the semi-randommethod which was developed by Kim [11] to determine the asymptotic behavior of Ramseynumbers r ( K , K k ). • It would be interesting to extend an upper bound in Theorem 1.2 to the general case when H and G are arbitrary graphs with sizes t and m and with no isolated vertices. We conjecturethat if t is fixed and m is sufficiently large then r ( H, G ) ≤ m O ( √ t ) . This estimate if true is tight up to a constant in the exponent, since the known lower bounds(see [16, 13]) on off-diagonal Ramsey numbers imply that r ( H, G ) ≥ m Ω( √ t ) when H and G are complete graphs with t and m edges respectively. In [3] it was proved that if H is agraph with chromatic number ℓ and maximum degree d ≥ ℓ then for all sufficiently large m , r ( H, K m ) ≤ m ℓd . Using this estimate it is easy to obtain the following partial result, whichshows that our conjecture holds if the chromatic number of H is a fixed constant. Proposition 3.2
Let H and G be two graphs with no isolated vertices such that the size of G is m , the size of H is t and H has chromatic number ℓ ≥ . Then there exists a constant c depending only on ℓ such that for sufficiently large m , r ( H, G ) ≤ m c √ t . Sketch of proof.
We use induction on t . Let v be the vertex of maximum degree in H . Since G has m edges, it has at most 2 m vertices. Therefore, if the maximum degree of H is at most6 √ t , it follows from the above cited estimate in [3] that r ( H, G ) ≤ m √ tℓ . Otherwise, the degreeof v in H is larger than 2 √ t . Delete it and denote H = H \ { v } . This graph has t ≤ t − √ t edges and √ t ≤ √ t − n = m √ tℓ and consider red-blue coloring of the edges of the complete graph K n . Since G has at most 2 m vertices, we can assume that there is no red K m in this coloring. Therefore, byTur´an’s theorem, there is a vertex x in K n , whose blue degree is at least n/ m ≫ m (2 √ t − ℓ ≥ m √ t ℓ . Let U be the set of blue neighbors of x . Clearly, this set contains no blue copy of H .Now we can use induction to conclude that it contains a red copy of G . (cid:3) References [1] M. Ajtai, J. Koml´os and E. Szemer´edi, A note on Ramsey numbers,
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