A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ which assigns at least $q$ colors to each $p$-clique. The problem of determining the minimum number of colors, $f(n,p,q)$, needed to give a $(p,q)$-coloring of the complete graph $K_n$ is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers $r_k(p)$. The best-known general upper bound on $f(n,p,q)$ was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where $p=q$ have been obtained only for $p\in\{4,5\}$, each of which was proved by giving a deterministic construction which combined a $(p,p-1)$-coloring using few colors with an algebraic coloring.
In this paper, we provide a framework for proving new upper bounds on $f(n,p,p)$ in the style of these earlier constructions. We characterize all colorings of $p$-cliques with $p-1$ colors which can appear in our modified version of the $(p,p-1)$-coloring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying $(p,p)$-colorings, which would otherwise make this problem intractable for large values of $p$. In addition, we generalize our algebraic coloring from the $p=5$ setting and use this to give improved upper bounds on $f(n,6,6)$ and $f(n,8,8)$.
We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdos distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs R(3,k).
The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erdős and Szekeres in 1935, asserts that the Ramsey number of the complete graph with $m$ edges is at most $2^{O(\sqrt{m})}$. Motivated by this estimate Erdős conjectured, more than a quarter century ago, that there is an absolute constant $c$ such that $r(G) \leq 2^{c\sqrt{…
Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erdos-Rado (which was the key to the Erdos-Rado proof), yield improved bounds on the a-hypergraph Ramsey numbers. We present all of these proofs and state explicit bounds for the 2-color case and the c-color case. We give a more detailed analysis of the…
Motivated by the Erdos-Szekeres convex polytope conjecture in $R^d$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $ n > k \geq 5$, what is the minimum integer $g_k(n)$ such that any $k$-uniform hypergraph on $g_k(n)$ vertices with the property that any set of $k + 1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_k(n) > 2^{cn^{k-4}}$, where $c = c(k)$.
Given integers $\ell,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell}$ with vertex set $v_1<v_2<\cdots < v_n$, and all edges of the form $v_iv_j$ where $|i-j|\le \ell$. The ramsey number $r(P_n^{\ell}, P_n^{\ell})$ is the minimum $N$ such that every 2-coloring of ${[N] \choose 2}$ results in a monochromatic copy of $P_n^{\ell}$. It is well-known that $r(P_n^1, P_n^1)=(n-1)^2+1$. For $\ell>1$, Balko-Cibulka-Král-Kynčl proved that $r(P_n^{\ell}, P_n^{\ell})< c_{\ell}n^{128 \ell}$ and asked for the growth rate for fixed $\ell$. When $\ell=2$, we improve this upper bound b…
In this paper we show that there exists a constant $C>0$ such that for any graph $G$ on $Ck\ln k$ vertices either $G$ or its complement $\bar{G}$ has an induced subgraph on $k$ vertices with minimum degree at least $\frac12(k-1)$. This affirmatively answers a question of Erdős and Pach from 1983.
We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for $k$-uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.
For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there is an absolute constant c such that R(G) = R(G,G) < 2c p m for any graph G with m edges and no isolated vertices. Recently this conjecture was proved by B. Sudakov. In this note, using the Sudakovs ideas we give an extension of his result and some interesting corollaries.
The Erdős-Gyárfás number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that all of its $p$-clique spans at least $q$ colors. In this paper we improve the best known upper bound on $f(n, p, q)$ for many fixed values of $p, q$ and large $n$. Our proof uses a randomized coloring process, which we analyze using the so-called differential equation method to establish dynamic concentration.
Given graphs $ F_1, F_2$ and $G$, we say that $G$ is Ramsey for $(F_1,F_2)$ and we write $G\rightarrow(F_1, F_2)$, if for every edge coloring of $G$ by red and blue, there is either a red copy of $F_1$ or a blue copy of $F_2$ in $G$. The size Ramsey number $\hat{r}(F_1, F_2)$ is defined as the minimum number of edges of a graph $G$ such that $G\rightarrow(F_1, F_2)$. This paper provides the exact value of $\hat{r}(F_1, F_2)$ for many pairs $(F_1, F_2)$ of star forests, giving a partial solution to a conjecture of Burr et al. (Ramsey-minimal graphs for multiple copies, Indagationes Mathematicae…
An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-Erdős-Sós conjecture states that for every fixed $k$ and $r$, every linear $r$-graph with $Ω(n^2)$ edges contains $k$ edges spanned by at most $(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete linear …
Fix positive integers $p$ and $q$ with $2 \leq q \leq {p \choose 2}$. An edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ different colors. The function $f(n, p, q)$ is the minimum number of colors that are needed for $K_n$ to have a $(p,q)$-coloring. This function was introduced by Erdős and Shelah about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that $f(n, p, p)$ is polynomial in $n$ and asked to determine the maximum $q$, depending on $p$, for which $f(n,p,q)$ is su…
Extending the concept of Ramsey numbers, Erd{\H o}s and Rogers introduced the following function. For given integers $2\le s<t$ let $$ f_{s,t}(n)=\min \{\max \{|W| : W\subseteq V(G) {and} G[W] {contains no} K_s\} \}, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this paper, we show that for every $s\ge 3$ there exist constants $c_1=c_1(s)$ and $c_2=c_2(s)$ such that $f_{s,s+1}(n) \le c_1 (\log n)^{c_2} \sqrt{n}$. This result is best possible up to a polylogarithmic factor. We also show for all $t-2 \geq s \geq 4$, there exists a constant $c_3$ such that $f_{s,t}…
In a seminal paper from 1983, Burr and Erdos started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving all but one of the Burr-Erdos problems.
For given simple graphs $G_1$ and $G_2$, the size Ramsey number $\hat{R}(G_1,G_2)$ is the smallest positive integer $m$, where there exists a graph $G$ with $m$ edges such that in any edge coloring of $G$ with two colors red and blue, there is either a red copy of $G_1$ or a blue copy of $G_2$. In 1981, Erdős and Faudree investigated the size Ramsey number $\hat{R}(K_n,tK_2)$, where $K_n$ is a complete graph on $n$ vertices and $tK_2$ is a matching of size $t$. They obtained the value of $\hat{R}(K_n,tK_2)$ when $n\geq 4t-1$ as well as for $t=2$ and asked for the behavior of these numbers when…
A classical and widely used lemma of Erdos and Szekeres asserts that for every n there exists N such that every N-term sequence a of real numbers contains an n-term increasing subsequence or an n-term nondecreasing subsequence; quantitatively, the smallest N with this property equals (n-1)^2+1. In the setting of the present paper, we express this lemma by saying that the set of predicates Phi={x_1<x_2,x_1\ge x_2}$ is Erdos-Szekeres with Ramsey function ES_Phi(n)=(n-1)^2+1. In general, we consider an arbitrary finite set Phi={Phi_1,...,Phi_m} of semialgebraic predicates, meaning that each Phi_j…
Let $Δ_s=R(K_3,K_s)-R(K_3,K_{s-1})$, where $R(G,H)$ is the Ramsey number of graphs $G$ and $H$ defined as the smallest $n$ such that any edge coloring of $K_n$ with two colors contains $G$ in the first color or $H$ in the second color. In 1980, Erdős and Sós posed some questions about the growth of $Δ_s$. The best known concrete bounds on $Δ_s$ are $3 \le Δ_s \le s$, and they have not improved since the stating of the problem. In this paper we present some constructions, which imply in particular that $R(K_3,K_s) \ge R(K_3,K_{s-1}-e) + 4$. This does not improve the lower bound of 3 on $Δ_s$, b…
A graph on $n$ vertices is said to be \emph{$C$-Ramsey} if every clique or independent set of the graph has size at most $C \log n$. The only known constructions of Ramsey graphs are probabilistic in nature, and it is generally believed that such graphs possess many of the same properties as dense random graphs. Here, we demonstrate one such property: for any fixed $C>0$, every $C$-Ramsey graph on $n$ vertices induces subgraphs of at least $n^{2-o(1)}$ distinct sizes. This near-optimal result is closely related to two unresolved conjectures, the first due to Erdős and McKay and the second due …
We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform version $r_k(k+1,n)$. This is the first improvement of the original exponential lower bound for $r_4(5,n)$ implicit in work of Erd\H os and Hajnal from 1972 and also improves the current best known bounds for larger $k$ due to the authors. Second, we prove an upper bound for the hypergraph Erd\H os-Rogers function $f^k_{k+1, k+2}(N)$ that is an iterated $(k-13)…
The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erdős in 1975 conjectured that for each positive integer d there is a constant c_d such that r(H) \leq c_dn for every d-degenerate graph H on n vertices. We show that for such graphs r(H) \leq 2^{c_d\sqrt{\log n}}n, improving on an earlier bound of Kostochka and Sudakov. We also study Ramsey numbers of random graphs, showi…
In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the…
Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of its $k$-subsets blue. Erdős and Hajnal proved in 1972 that for fixed $2\le t \le k$, there are positive constants $c_1$ and $c_2$ such that $$ 2^{c_1 n} < g_k(t, n) < twr_{t-1} (n^{c_2}),$$ where $twr_{t-1}$ is a tower of 2's of height $t-2$. They conjectured that the tower growth rate in the upper bound is correct. Despite decades of work on closely related and s…
Let H be a k-uniform hypergraph whose vertices are the integers 1,...,N. We say that H contains a monotone path of length n if there are x_1 < x_2 < ... < x_{n+k-1} so that H contains all n edges of the form {x_i,x_{i+1},...,x_{i+k-1}}. Let N_k(q,n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of N_k(q,n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdos and Szekeres, the problem of bounding N_k(q,n) for arbitrary k and…
In this paper we prove several results on Ramsey numbers $R(H,F)$ for a fixed graph $H$ and a large graph $F$, in particular for $F = K_n$. These results extend earlier work of Erdős, Faudree, Rousseau and Schelp and of Balister, Schelp and Simonovits on so-called Ramsey size-linear graphs. Among others, we show that if $H$ is a subdivision of $K_4$ with at least $6$ vertices, then $R(H,F) = O(v(F) + e(F))$ for every graph $F$. We also conjecture that if $H$ is a connected graph with $e(H) - v(H) \leq \binom{k+1}{2} - 2$, then $R(H,K_n) = O(n^k)$. The case $k=2$ was proved by Erdős, Faudree, R…
We find the Ramsey number of a cycle vs. a complete graph when the order of the cycle is at least 4 times as large as the order of the complete graph. This partially confirms a conjecture of Erdős, Faudree, Rousseau, and Schelp made in 1978.
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
For two graph H and G, the Ramsey number r(H, G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph K_n on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdos and Harary, in this note we study how the Ramsey number r(K_s, G) depends on the size of the graph G. For s \geq 3, we prove that for every G with m edges, r(K_s,G) \geq c (m/\log m)^{\frac{s+1}{s+3}} for some positive constant c depending only on s. This lower bound improves an earlier result of Erdos, Faudree, Rousseau, and Schelp, and is tight up …
In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1\le r\le e(H)$ such graphs occur naturally at intermediate steps in the synthesis of a $2$-colour Ramsey graph $G_1\longrightarrow H$. (The corresponding notion of Ramsey-type numbers was introduced by Erdös, Hajnal and Rado in 1965 and subsequently studied by Erdös and Szemerédi in 1972). For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and establish Ramsey-infiniteness. From the structural point of view, …
The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erdős, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq C\frac {\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon >0$ and $n>…
In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Jo…
The study of problems concerning subdivisions of graphs has a rich history in extremal combinatorics. Confirming a conjecture of Burr and Erdős, Alon proved in 1994 that subdivided graphs have linear Ramsey numbers. Later, Alon, Krivelevich and Sudakov showed that every $n$-vertex graph with at least $\varepsilon n^2$ edges contains a $1$-subdivision of the complete graph on $c_{\varepsilon}\sqrt{n}$ vertices, resolving another old conjecture of Erdős. In this paper we consider the directed analogue of these problems and show that every tournament on at least $(2+o(1))k^2$ vertices contains th…
This paper deals with the minimum number $m_H(r)$ of edges in an $H$-free graph with the chromatic number more than $r$. We show how bounds on Ramsey and Turán numbers imply bounds on $m_H(r)$.
Let r be an integer, f(n) a function, and H a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi, the r-Ramsey-Turán number of H, RT_r(n, H, f(n)), is defined to be the maximum number of edges in an n-vertex, H-free graph G with α_r(G) <= f(n) where α_r(G) denotes the K_r-independence number of G. In this note, using isoperimetric properties of the high dimensional unit sphere, we construct graphs providing lower bounds for RT_r(n,K_{r+s},o(n)) for every 2 <= s <= r. These constructions are sharp for an infinite family of pairs of r and s. The only previous sharp construction was by Bollob…
The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs $G$ for which the value of $r(G)$ is known exactly. One such family consists of large vertex disjoint unions of a fixed graph $H$, we denote such a graph, consisting of $n$ copies of $H$ by $nH$. This classical result was proved by Burr, Erdős and Spencer in 19…
The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3 and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a …
A graph is $d$-degenerate if all its subgraphs have a vertex of degree at most $d$. We prove that there exists a constant $c$ such that for all natural numbers $d$ and $r$, every $d$-degenerate graph $H$ of chromatic number $r$ with $|V(H)| \ge 2^{d^22^{cr}}$ has Ramsey number at most $2^{d2^{cr}} |V(H)|$. This solves a conjecture of Burr and Erdős from 1973.
We show that, for $n$ large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic $K_t$s in any two-colouring of the edges of $K_n$, where $C \approx 2.18$ is an explicitly defined constant. The old lower bound, due to Erdős \cite{E62}, and based upon the standard bounds for Ramsey's theorem, is \[\frac{n^t}{4^{(1+o(1))t^2}}.\]
Let $r_k(C_{2m+1})$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge2$, \[r_k(C_{2m+1})<c^{k}\sqrt{k!}\] for all sufficiently large $k$, where $c=c(m)>0$ is a constant. This improves an old result by Bondy and Erdős (Ramsey numbers for cycles in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).
The multicolor Ramsey number problem asks, for each pair of natural numbers $\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. Graphs useful for this construction are exactly those relevant for a problem of Erdős on graphs with no large cliques and few large independent sets. We also make some basic observations about thi…
The following relaxation of the classical problem of determining Ramsey number of a fixed graph has first been proposed by Erdos, Hajnal and Rado over 50 years ago. Given a graph $G$ and an integer $t \geq 2$ determine the minimum number $N$ such that in any $t$-coloured complete graph on $N$ vertices there is a copy of $G$ using only edges of some $t-1$ colours. We determine the answer precisely when $G$ is a path.
The first application of Szemerédi's powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible indepe…
The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique of size $k$. The case $s = 1$ is the usual $r$-colour Ramsey number, and the case $s = r - 1$ was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general $s$ were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that $R_{r,s}(k) = 2^{Θ(kr)}$ if $s/r$ is bounde…
The Ramsey number $r_k(s,n)$ is the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ contains a red set of size $s$ or a blue set of size $n$, where a set is red (blue) if all of its $k$-subsets are red (blue). A $k$-uniform \emph{tight path} of size $s$, denoted by $P_{s}$, is a set of $s$ vertices $v_1 < \cdots < v_{s}$ in $\mathbb{Z}$, and all $s-k+1$ edges of the form $\{v_j,v_{j+1},\ldots, v_{j + k -1}\}$. Let $r_k(P_s, n)$ be the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ results in a red $P_{s}$ or a blue …
We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on…
Ramsey number of a cycle versus a graph of a given size
KU Leuven Campus Kulak-Kortrijk（荷语鲁汶大学库拉-科特赖克校区）；HUN-REN, Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络，阿尔弗雷德·雷尼数学研究所）；ETH Zürich（苏黎世联邦理工学院）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；University College London（伦敦大学学院）
