Given a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ if $V(G)\subset V(\mathcal{H})$ and there is a bijection $f:E(G)\rightarrow E(\mathcal{H})$ such that for any edge $e$ of $G$ we have $e\subset f(e)$. We study Ramsey problems for Berge copies of graphs, i.e. the smallest number of vertices of a complete $r$-uniform hypergraph, such that if we color the hyperedges with $c$ colors, there is a monochromatic Berge copy of $G$.
We obtain a couple results regarding these problems. In particular, we determine for which $r$ and $c$ the Ramsey number can be super-linear. We also show a new way to obtain lower bounds, and improve the general lower bounds by a large margin. In the specific case $G=K_n$ and $r=2c-1$, we obtain an upper bound that is sharp besides a constant term, improving earlier results.
In this paper, for small uniformities, we determine the order of magnitude of the multicolor Ramsey numbers for Berge cycles of length $4$, $5$, $6$, $7$, $10$, or $11$. Our result follows from a more general setup which can be applied to other hypergraph Ramsey problems. Using this, we additionally determine the order of magnitude of the multicolor Ramsey number for Berge-$K_{a,b}$ for certain $a$, $b$, and uniformities.
For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $Φ:E(G)\longrightarrow E( \mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq Φ(e)$. We denote by $\mathcal{B}^rG$, the family of $r$-uniform Berge-$G$ hypergraphs. For families $\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t$ of $r$-uniform hypergraphs, the Ramsey number $R(\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t)$ is the smallest integer $n$ such that in every $t$-hyperedge coloring of $\mathcal{K}_{n}^r$ there is a monochromatic copy of a hypergraph in $\mathcal{H}…
For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Be…
In this paper, we consider an extension of cycle-complete graph Ramsey numbers to Berge cycles in hypergraphs: for $k \geq 2$, a {\em non-trivial Berge $k$-cycle} is a family of sets $e_1,e_2,\dots,e_k$ such that $e_1 \cap e_2, e_2 \cap e_3,\dots,e_k \cap e_1$ has a system of distinct representatives and $e_1 \cap e_2 \cap \dots \cap e_k = \emptyset$. In the case that all the sets $e_i$ have size three, let $\mathcal{B}_k$ denotes the family of all non-trivial Berge $k$-cycles. The {\em Ramsey numbers} $R(t,\mathcal{B}_k)$ denote the minimum $n$ such that every $n$-vertex $3$-uniform hypergrap…
For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection f from E(G) to E(H) such that for each e in E(G), e is a subset of f(e). The set of all Berge-G hypergraphs is denoted B(G). For integers k>1, r>1, and a graph G, let the Ramsey number R_r(B(G), k) be the smallest integer n such that no matter how the edges of a complete r-uniform n-vertex hypergraph are colored with k colors, there is a copy of a monochromatic Berge-G subhypergraph. Furthermore, let R(B(G),k) be the smallest integer n such that no matter how all subsets an n-element set are colored with k colors, th…
For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of $\mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the \emph{cover Ramsey number}, denoted as $\hat{R}^R(BG_1…
For a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $G$ (or a Berge-$G$ in short), if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We denote the family of $r$-uniform hypergraphs that are Berge copies of $G$ by $B^rG$. For families of $r$-uniform hypergraphs $\mathbf{H}$ and $\mathbf{H}'$, we denote by $R(\mathbf{H},\mathbf{H}')$ the smallest number $n$ such that in any blue-red coloring of $\mathcal{K}_n^r$ (the complete $r$-uniform hypergraph on $n$ vertices) there is a monochromatic blue copy of a hypergrap…
The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$ colors, there is a copy of $\mathcal{H}$ whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large $n$, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtai…
A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Grósz, Methuku and Tompkins in 2020 showed that for any graph $F$, there is an integer $r_0=r_0(F)$, such that for any $r\ge r_0$, any $r$-uniform hypergraph without a Berge copy of $F$ has $o(n^2)$ hyperedges. The smallest such $r_0$ is called the uniformity threshold of $F$ and is denoted by $th(F)$. They showed that $th(F)\le R(F,F')$, where $R$ denotes the off-diagonal Ramsey number and $F'$ is any graph obtained form $F$ by deleting an edge. We improve this bound to $th(F)\le R(K_{χ(F)},F')$, and use the …
For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from $F$ by replacing each edge of $F$ with a hyperedge containing it. A hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity $\sum_{h \in E(\mathcal{H})} |h|$. Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\mathcal{H}$ has size at least the Ramsey number of $F$ and at most $o(n)$, the weight of $\mathcal{H}$ is $o(n^2)$. This result i…
The degree anti-Ramsey number $AR_d(H)$ of a graph $H$ is the smallest integer $k$ for which there exists a graph $G$ with maximum degree at most $k$ such that any proper edge colouring of $G$ yields a rainbow copy of $H$. In this paper we prove a general upper bound on degree anti-Ramsey numbers, determine the precise value of the degree anti-Ramsey number of any forest, and prove an upper bound on the degree anti-Ramsey numbers of cycles of any length which is best possible up to a multiplicative factor of $2$. Our proofs involve a variety of tools, including a classical result of Bollobás c…
Given a hypergraph $\mathcal{H}$ and a graph $G$, we say that $\mathcal{H}$ is a \textit{Berge}-$G$ if there is a bijection between the hyperedges of $\mathcal{H}$ and the edges of $G$ such that each hyperedge contains its image. We denote by $ex_k(n,\text{Berge-}F)$ the largest number of hyperedges in a $k$-uniform Berge-$F$-free graph. Let $ex(n,H,F)$ denote the largest number of copies of $H$ in $n$-vertex $F$-free graphs. It is known that $ex(n,K_k,F)\le ex_k(n,\text{Berge-}F)\le ex(n,K_k,F)+ex(n,F)$, thus if $χ(F)>r$, then $ex_k(n,\text{Berge-}F)=(1+o(1)) ex(n,K_k,F)$. We conjecture that …
Let $F$ be a graph. We say that a hypergraph $H$ is a {\it Berge}-$F$ if there is a bijection $f : E(F) \rightarrow E(H )$ such that $e \subseteq f(e)$ for every $e \in E(F)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$F$ is denoted $\ex_r(n,\textrm{Berge-}F)$. In this paper we establish new upper and lower bounds on $\ex_r(n,\textrm{Berge-}F)$ for general graphs $F$, and investigate connections between $\ex_r(n,\textrm{Berge-}F)$ and other recently studied extremal functions…
For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$ In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results. I…
We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycles. Next we give an upper bound for the number of hyperedges in a hypergraph with bounded circumference. Both results are best possible in infinitely many cases.
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…
A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class, or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions. We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an $O(n^9)$-…
Fix a hypergraph $\mathcal{F}$. A hypergraph $\mathcal{H}$ is called a {\it Berge copy of $\mathcal{F}$} or {\it Berge-$\mathcal{F}$} if we can choose a subset of each hyperedge of $\mathcal{H}$ to obtain a copy of $\mathcal{F}$. A hypergraph $\mathcal{H}$ is {\it Berge-$\mathcal{F}$-free} if it does not contain a subhypergraph which is Berge copy of $\mathcal{F}$. This is a generalization of the usual, graph based Berge hypergraphs, where $\mathcal{F}$ is a graph. In this paper, we study extremal properties of hypergraph based Berge hypergraphs and generalize several results from the graph ba…
We reveal a connection between the incompressibility method and the Lovasz local lemma in the context of Ramsey theory. We obtain bounds by repeatedly encoding objects of interest and thereby compressing strings. The method is demonstrated on the example of van der Waerden numbers. It applies to lower bounds of Ramsey numbers, large transitive subtournaments and other Ramsey phenomena as well.
A \emph{simple} matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a \emph{Berge hypergraph} if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $||A||$ denote the number of columns in $A$, we define the extremal function $Bh(m,{ F})=\max\{||A||\,:\, A \hbox{ is }m\hbox{-rowed simple matrix with no Berge hypergraph }F\}$. We determine the asymptotics of $Bh(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determini…
In this paper we study the maximum number of hyperedges which may be in an $r$-uniform hypergraph under the restriction that no pair of vertices has more than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the even-cycle problem asking for $\mathrm{ex}(n, C_{2k})$. We extend results of Füredi and Simonovits and of Conlon, who studied the problem when $r=2$. In particular, we show that for fixed $k$ and $r$, there is a constant $t$ such that the maximum number of edges can be determined in order of magnitude.
A hypergraph $H=(V(H), E(H))$ is a Berge copy of a graph $F$, if $V(F)\subset V(H)$ and there is a bijection $f:E(F)\rightarrow E(H)$ such that for any $e\in E(F)$ we have $e\subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain any Berge copies of $F$. We address the saturation problem concerning Berge-$F$-free hypergraphs, i.e., what is the minimum number $sat_r(n,F)$ of hyperedges in an $r$-uniform Berge-$F$-free hypergraph $H$ with the property that adding any new hyperedge to $H$ creates a Berge copy of $F$. We prove that $sat_r(n,F)$ grows linearly in $n$ if $F$ is either c…
In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph $\mathcal{H}$ is a $t$-heavy copy of a graph $F$ if there is a copy of $F$ on its vertex set such that each edge of $F$ is contained in at least $t$ hyperedges of $\mathcal{H}$. $\mathcal{H}$ is a $t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$ hyperedges are distinct. We extend known upper bounds on the Turán number of Be…
The anti-Ramsey number $Ar(G,H)$ is the maximum number of colors in an edge-coloring of $G$ with no rainbow copy of $H$. In this paper, we determine the exact anti-Ramsey number in the generalized Petersen graph $P_{n,k}$ for cycles $C_d$, where $1\leq k\leq \lfloor \frac{n-1}{2} \rfloor$ and $5\le d \le 6$. We also give an algorithm to obtain the upper bound or lower bound of anti-Ramsey number.
For a graph G, a hypergraph H is called Berge-G if there is a hypergraph H', isomorphic to H, containing all vertices of G, so that e is contained in f(e) for each edge e of G, where f is a bijection between E(G) and E(H'). The set of all Berge-G hypergraphs is denoted B(G). A hypergraph H is called Berge-G saturated if it does not contain any subhypergraph from B(G), but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Each Berge-G saturated hypergraph has at least |E(G)|-1 hyperedges. We show that for each graph G that is not a certain star and for any n at leas…
The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments and open problems.
We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges $|E_{RB}|$ is given. The aim is to find the maximal size $f$ of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on $n$ vertices such that the number of bi-oriented edges $|E_{bi}|$ is given. The aim is to bound the size $F$ of the maximal transitive subtournament that is guaran…
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.
We introduce and study a new type of Ramsey-Turan problems, a typical example of which is the following one: let c>0 and G be a graph of sufficiently large order n with minimum degree >3n/4. If the edges of G are colored in blue or red, then for every 3<k<(1/8-c)n, there exists a monochromatic cycle of length k.
In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that finds the desired colouring with high probability. Our framework allows to reduce the probabilistic problem of whether the Ramsey property at hand holds for random (hyper)graphs with edge probability $p$ to a deterministic question of whether there exists a finite graph that forms an obstruction. In the second part of the paper we apply this framework to addres…
We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.
In 1981, Alspach conjectured that the complete graph $ K_{n} $ could be decomposed into cycles of arbitrary lengths, provided that the obvious necessary conditions would hold. This conjecture was proved completely by Bryant, Horsley and Pettersson in 2014. Moreover, in 1983, Tarsi conjectured that the obvious necessary conditions for packing pairwise edge-disjoint paths of arbitrary lengths in the complete multigraphs were also sufficient. The conjecture was confirmed by Bryant in 2010. In this paper, we investigate an analogous problem as the decomposition of the complete uniform multi-hyperg…
A classical conjecture of Erdős and Sós asks to determine the Turán number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, for all $k$ and $r$, with $r \ge k (k-2)$, we show that any $r$-uniform hypergraph $\mathcal{H}$ with more than $\frac{n(k-1)}{r+1}$ hyperedges contains a Berge copy of any tree with $k$ edges different from the $k$-edge star. This bound is sharp when $r+1$ divides $n$ and for such values of $n$ we determine the extremal hypergraphs.
We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We consider both weak and strong (non-induced and induced) versions of this problem. We also investigate related problems on (partial) $k$-colorings of $B_n$ that do not admit rainbow antichains of size $k$.
The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $Γ_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r) \leq r^{\binom{r+1}{2}} - \left( 1/4 - o(1) \right) r^{\binom{r}{2}+1} $, slightly improving the currently best known upper bound due to Gyárfás.
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).
Let $n,r,k,s$ be positive integers with $n,k\ge 2$. The generalized Ramsey number $R(n,r;k,s)$ is the smallest positive integer $p$ such that for every graph $G$ of order $p$, either $G$ contains a subgraph induced by $n$ vertices with at most $r-1$ edges, or the complement $\bar G$ of $G$ contains a subgraph induced by $k$ vertices with at most $s-1$ edges. In this paper we completely determine $R(n,n(n-1)/2-r;k,1)$ for $n\ge 4$ and $r\le n-2$, and pose several conjectures on Ramsey numbers.
In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem.
It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it is improved to 13 <= N^* <= N.
The size Ramsey number of a graph $H$ is defined as the minimum number of edges in a graph $G$ such that there is a monochromatic copy of $H$ in every two-coloring of $E(G)$. The size Ramsey number was introduced by Erdős, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs (obtained by adding many common neighbors to the vertices of a clique), and starburst graphs (obtained by adding many pendant edges to each vertex …
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 $c = c(k)$.
If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently large, contains a monochromatic copy of S. Ramsey's theorem tells us that for any r \geq 1 every 2-coloring of a sufficiently large r-uniform hypergraph will contain a large monochromatic clique (a complete subhypergraph): hence any set of vertices from Q_d that all have the same weight is Ramsey. A natural question to ask is: which sets S corresponding to u…
We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed by Graham and Rothschild and they gave an upper bound: $n\le F(F(F(F(F(F(F(12)))))))$, where $F(m) = 2\uparrow^m3$. In 2014 Lavrov, Lee and Mackey greatly improved this result by giving upper bound $n< 2\uparrow\uparrow\uparrow 6 < F(5)$. In this paper we revisit their estimates and reduce upper bound to $n< 2\uparrow\uparrow\uparrow 5$
Starting from an innocent Ramsey-theoretic question regarding directed paths in tournaments, we discover a series of rich and surprising connections that lead into the theory around a fundamental problem in Combinatorics: the Ruzsa-Szemerédi induced matching problem. Using these relationships, we prove that every coloring of the edges of the transitive $n$-vertex tournament using three colors contains a directed path of length at least $\sqrt{n} \cdot e^{\log^* n}$ which entirely avoids some color. We also expose connections to a family of constructions for Ramsey tournaments, and introduce an…
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.
A graph on 5 vertices consisting of 2 copies of the cycle graph C3 sharing a common vertex is called the Butterfly graph (B). The smallest natural number s such that any two-colouring (say red and blue) of the edges of Kj*s has a copy of a red B or a blue G is called the multipartite Ramsey number of Butterfly graph versus G. This number is denoted by mj(B,G). In this paper we find exact the values for mj(B,G) when and G represents any connected proper subgraph of K4 with at least one edge.
