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Assume that $R_1,R_2,\dots,R_t$ are disjoint parallel lines in the plane. A $t$-interval (or $t$-track interval) is a set that can be written as the union of $t$ closed intervals, each on a different line. It is known that pairwise intersecting $2$-intervals can be pierced by two points, one from each line. However, it is not true that every set of pairwise intersecting $3$-intervals can be pierced by three points, one from each line. For $k\ge 3$, Kaiser and Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by $t$ points, one from each line. Our main result provides an asymptotic answer: in any set $S_1,\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least $\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove this in a more general form, replacing intervals by subtrees of a tree. This leads to questions and results on covering vertices of edge-colored complete graphs by vertices of monochromatic cliques having distinct colors, where the colorings are chordal, or more generally induced $C_4$-free graphs. For instance, we show that if the edges of a complete graph $K_n$ are colored with red or blue so that both color classes are induced $C_4$-free, then at least ${4n\over 5}$ vertices can be covered by a red and a blue clique, and this is best possible. We conclude by pointing to new Ramsey-type problems emerging from these restricted colorings., Comment: 15 pages, 3 figures

Two independent edges in ordered graphs can be nested, crossing or separated. These relations define six types of subgraphs, depending on which relations are forbidden. We refine a remark by Erd\H{o}s and Rado that every 2-coloring of the edges of a complete graph contains a monochromatic spanning tree. We show that forbidding one relation we always have a monochromatic (non-nested, non-crossing, non-separated) spanning tree in a 2-edge-colored ordered complete graph. On the other hand, if two relations are forbidden, then it is possible that we have monochromatic (nested, separated, crossing) subtrees of size only half the number of vertices. The existence of a monochromatic non-nested spanning tree in 2-colorings of ordered complete graphs verifies a more general conjecture for twisted drawings. Our second subject is to refine the Ramsey number of matchings for ordered complete graphs. Cockayne and Lorimer proved that for given positive integers t, n, m=(t-1)(n-1)+2n is the smallest integer with the following property: every t-coloring of the edges of a complete graph Km contains a monochromatic matching with n edges. We conjecture a strengthening: t-colored ordered complete graphs on m vertices contain monochromatic non-nested and also non-separated matchings with n edges. We prove this conjecture for some special cases. (i) Every t-colored ordered complete graph on t+3 vertices contains a monochromatic non-nested matching of size two. (ii) Every 2-colored ordered complete graph on 3n-1 vertices contains a monochromatic non-separated matching of size n. For nested, separated, and crossing matchings the situation is different. The smallest m ensuring a monochromatic matching of size n in every t-coloring is 2(t(n-1))+1) in the first two cases and one less in the third case.

A \textit{linear $3$-graph}, $H = (V, E)$, is a set, $V$, of vertices together with a set, $E$, of $3$-element subsets of $V$, called edges, so that any two distinct edges intersect in at most one vertex. The linear Tur\'an number, ${\rm ex}(n,F)$, is the maximum number of edges in a linear $3$-graph $H$ with $n$ vertices containing no copy of $F$. We focus here on the \textit{crown}, $C$, which consists of three pairwise disjoint edges (jewels) and a fourth edge (base) which intersects all of the jewels. Our main result is that every linear $3$-graph with minimum degree at least $4$ contains a crown. This is not true if $4$ is replaced by $3$. In fact the known bounds of the Tur\'an number are \[ 6 \left\lfloor{\frac{n - 3}{4}}\right\rfloor \leq {\rm ex}(n, C) \leq 2n, \] and in the construction providing the lower bound all but three vertices have degree $3$. We conjecture that ${\rm ex}(n, C) \sim \frac{3n}{2}$ but even if this were known it would not imply our main result. Our second result is a step towards a possible proof of ${\rm ex}(n,C) \leq \frac{3n}{2}$ (i.e., determining it within a constant error). We show that a minimal counterexample to this statement must contain certain configurations with $9$ edges and we conjecture that all of them lead to contradiction., Comment: 10 pages, 5 figures, comments welcome

A {\em special four-cycle } $F$ in a triple system consists of four triples {\em inducing } a $C_4$. This means that $F$ has four special vertices $v_1,v_2,v_3,v_4$ and four triples in the form $w_iv_iv_{i+1}$ (indices are understood $\pmod 4$) where the $w_j$s are not necessarily distinct but disjoint from $\{v_1,v_2,v_3,v_4\}$. There are seven non-isomorphic special four-cycles, their family is denoted by $\cal{F}$. Our main result implies that the Tur\'an number $\text{ex}(n,{\cal{F}})=\Theta(n^{3/2})$. In fact, we prove more, $\text{ex}(n,\{F_1,F_2,F_3\})=\Theta(n^{3/2})$, where the $F_i$-s are specific members of $\cal{F}$. This extends previous bounds for the Tur\'an number of triple systems containing no Berge four cycles. We also study $\text{ex}(n,{\cal{A}})$ for all ${\cal{A}}\subseteq {\cal{F}}$. For 16 choices of $\cal{A}$ we show that $\text{ex}(n,{\cal{A}})=\Theta(n^{3/2})$, for 92 choices of $\cal{A}$ we find that $\text{ex}(n,{\cal{A}})=\Theta(n^2)$ and the other 18 cases remain unsolved., Comment: 7 figures

A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most $f(d)$ monochromatic $d$-dominating subgraphs. Note that the existence of $f(d)$ is not obvious -- indeed, the question which motivated this paper was simply to determine whether $f(d)$ is bounded, even for $d=2$. We answer this question affirmatively for all $d\geq 2$, proving $4\leq f(2)\le 8$ and $2d\leq f(d)\le 2d\left(\frac{d^{d}-1}{d-1}\right)$ for all $d\ge 3$. We also give an example to show that there is no analogous bound for more than two colors. Our result provides a positive answer to a question regarding an infinite analogue of the Burr-Erd\H{o}s conjecture on the Ramsey numbers of $d$-degenerate graphs. Moreover, a special case of our result is related to properties of $d$-paradoxical tournaments., Comment: 7 pages

In this paper we study Tur\'an and Ramsey numbers in linear triple systems, defined as $3$-uniform hypergraphs in which any two triples intersect in at most one vertex. A famous result of Ruzsa and Szemer\'edi is that for any fixed $c>0$ and large enough $n$ the following Tur\'an-type theorem holds. If a linear triple system on $n$ vertices has at least $cn^2$ edges then it contains a {\em triangle}: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called {\em $s$-configurations}. The main tool is a generalization of the induced matching lemma from $aba$-patterns to more general ones. We slightly generalize $s$-configurations to {\em extended $s$-configurations}. For these we cannot prove the corresponding Tur\'an-type theorem, but we prove that they have the weaker, Ramsey property: they can be found in any $t$-coloring of the blocks of any sufficiently large Steiner triple system. Using this, we show that all unavoidable configurations with at most 5 blocks, except possibly the ones containing the sail $C_{15}$ (configuration with blocks 123, 345, 561 and 147), are $t$-Ramsey for any $t\geq 1$. The most interesting one among them is the {\em wicket}, $D_4$, formed by three rows and two columns of a $3\times 3$ point matrix. In fact, the wicket is $1$-Ramsey in a very strong sense: all Steiner triple systems except the Fano plane must contain a wicket.

The notion of cross intersecting set pair system of size $m$, $\Big(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m\Big)$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne\emptyset$, was introduced by Bollob\'as and it became an important tool of extremal combinatorics. His classical result states that $m\le {a+b\choose a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$. Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is -- at least $5^{n/2}$ for $n$ even, $a=b=n$, -- equal to $\bigl(\lfloor\frac{n}{2}\rfloor+1\bigr)\bigl(\lceil\frac{n}{2}\rceil+1\bigr)$ if $a=2$ and $b=n\ge 4$, -- at most $|\cup_{i=1}^m A_i|$, -- asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ($|A_i\cap A_j|\le 1$ for $i\ne j$), -- asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.

A path-matching of order $p$ is a vertex disjoint union of nontrivial paths spanning $p$ vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers $r, p_1, \dots, p_r$, define $R^{PM}(p_1, \dots, p_r)$ to be the smallest integer $n$ such that in any $r$-coloring of the edges of $K_n$ there exists a path-matching of color $i$ and order at least $p_i$ for some $i\in [r]$. Our main result is that for $r\geq 2$ and $p_1\geq \dots\geq p_r\geq 2$, if $p_1\geq 2r-2$, then \[R^{PM}(p_1, \dots, p_r)= p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil.\] Perhaps surprisingly, we show that when $p_1<2r-2$, it is possible that $R^{PM}(p_1, \dots, p_r)$ is larger than $p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil$, but in any case we determine the correct value to within a constant (depending on $r$); i.e. \[p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil \leq R^{PM}(p_1, \dots, p_r)\leq \left\lceil p_1-\frac{r}{3}+\sum_{i=2}^r\frac{p_i}{3}\right\rceil.\] As a corollary we get that in every $r$-coloring of $K_n$ there is a monochromatic path-matching of order at least $3\left\lfloor\frac{n}{r+2}\right\rfloor$, which is essentially best possible. We also determine $R^{PM}(p_1, \dots, p_r)$ in all cases when the number of colors is at most 4. The proof of the main result uses a minimax theorem for path-matchings derived from a result of Las Vergnas (extending Tutte's 1-factor theorem) to show that the value of $R^{PM}(p_1, \dots, p_r)$ depends on the block sizes in covering designs (which can be also formulated in terms of monochromatic 1-cores in colored complete graphs). Then we obtain the result above by giving estimates on the block sizes in covering designs in the arbitrary (non-uniform) case., Comment: 12 pages, minor typos fixed, final version

Let $G=(V,E)$ be a $\tau$-critical graph with $\tau(G)=t$. Erd\H{o}s and Gallai proved that $|V|\leq 2t$ and the bound $|E|\leq {t+1\choose 2}$ was obtained by Erd\H{o}s, Hajnal and Moon. We give here the sharp combined bound $|E|+|V|\leq {t+2\choose 2}$ and find all graphs with equality.

A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$? In this paper, we give several sufficient conditions and one necessary condition to guarantee a positive answer to the above question. In particular, we prove the existence of a Gallai-coloring if $e_1-e_k\le 1$ and $k \le \lfloor n/2\rfloor$. We prove that for any integer $k\ge 3$ there is a (unique) integer $g(k)$ with the following property: there exists a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$ for every $e_1\le\dots \le e_k$ satisfying $\sum_{i=1}^ke_i={n\choose 2}$, if and only if $n\ge g(k)$. We show that $g(3)=5$, $g(4)=8$, and $2k-2\le g(k)\le 8k^2+1$ for every $k\ge 3$.

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$ such that $A\cap B$, $A\cap C$, and $B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$., Comment: 12 pages

A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples $\{e_{ij}: 1\le i< j \le 4\}$ such that $\{v_i,v_j\}\subset e_{ij}$ for every $1\le i

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, there is a monochromatic copy of a Berge-G hypergraph. We give an upper bound for R_r(B(G),k) in terms of graph Ramsey numbers. In particular, we prove that when G becomes acyclic after removing some vertex, R_r(B(G),k)\le 4k|V(G)|+r-2, in contrast with classical multicolor Ramsey numbers. When G is a triangle or a K_4, we find sharper bounds and some exact results and determine some `small' Ramsey numbers: k/2 - o(k) < R_3(B(K_3)), k) < 3k/4+ o(k), For any odd integer t\neq 3, R(B(K_3),2^t-1)=t+2, 2^{ck} < R_3(B(K_4),k) < e(1+o(1))(k-1)k!, R_3(B(K_3),2)=R_3(B(K_3),3)=5, R_3(B(K_3),4)=6, R_3(B(K_3),5)=7, R_3(B(K_3),6)=8, R_3(B(K_3,8)=9, R_3(B(K_4),2)=6., Comment: 11 pages

It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2n\over r}$ vertices. It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every $r$-coloring of $K_{n,n}$ there is a monochromatic component that meets both sides in at least $n/r$ vertices? Over forty years ago, Gy\'arf\'as and Lehel and independently Faudree and Schelp proved that any $2$-colored $K_{n,n}$ contains a monochromatic $P_n$. Very recently, Buci\'c, Letzter and Sudakov proved that every $3$-colored $K_{n,n}$ contains a monochromatic connected matching (a matching whose edges are in the same connected component) of size $\lceil n/3 \rceil$. So the answer is strongly "yes" for $1\leq r\leq 3$. We provide a short proof of (a non-symmetric version of) the original question for $1\leq r\leq 3$; that is, every $r$-coloring of $K_{m,n}$ has a monochromatic component that meets each side in a $1/r$ proportion of its part size. Then, somewhat surprisingly, we show that the answer to the question is "no" for all $r\ge 4$. For instance, there are $4$-colorings of $K_{n,n}$ where the largest balanced monochromatic component has $n/5$ vertices in both partite classes (instead of $n/4$). Our constructions are based on lower bounds for the $r$-color bipartite Ramsey number of $P_4$, denoted $f(r)$, which is the smallest integer $\ell$ such that in every $r$-coloring of the edges of $K_{\ell,\ell}$ there is a monochromatic path on four vertices. Furthermore, combined with earlier results, we determine $f(r)$ for every value of $r$., Comment: 9 pages, 2 figures, to appear in Journal of Combinatorics

In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\le k r$. This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property. Our other construction keeps $n-r$ fixed; in this case we show that for any fixed $k\ge 2$ there can only be finitely many examples. We give a range for $n$ where these hypergraphs exist. For $n-r=1$ the range is completely determined: $k+1\le n \le {(k+2)^2\over 4}$. For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

A hypergraph is linear if any two edges intersect in at most one vertex. For a fixed $k$-uniform family ${\cal{F}}$ of hypergraphs, the linear Tur\'an number ${\rm ex}_{\rm lin}(n,{\cal{F}})$ is the maximum number of edges in a $k$-uniform linear hypergraph $\mathcal H$ on $n$ vertices that does not contain any member of ${\cal{F}}$ as a subhypergraph. For $k\ge 2$ the $k$-fan $F^k$ is the $k$-uniform linear hypergraph having $k$ edges $f_1,\dots,f_k$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. We prove the following extension of Mantel's theorem $${\rm ex}_{\rm lin}(n,F^k)\le {n^2 / k^2}.$$ Moreover, $|{\mathcal H}|=n^2/k^2$ holds if and only if $n\equiv 0\pmod k$ and $\mathcal H$ is a transversal design on $n$ points with $k$ groups. We also study ${\rm ex}_{\rm lin}(n,{\cal{F}})$ where $\cal{F}$ is any subset of the three linear triple systems with four triples on at most seven points., Comment: 9 pages, 1 figure

A well-known special case of a conjecture attributed to Ryser states that k-partite intersecting hypergraphs have transversals of at most k-1 vertices. An equivalent form was formulated by Gy\'arf\'as: if the edges of a complete graph K are colored with k colors then the vertex set of K can be covered by at most k-1 sets, each connected in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: Z. Kir\'aly proved that in every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >= 3), the vertex set of K^r can be covered by at most $\lceil k/r \rceil$ sets, each connected in some color. Here we investigate the analogue problem for complete r-uniform r-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of any color used in the coloring. We propose the following analogue of Ryser conjecture. In every spanning (r+t)-coloring of the edges of a complete r-uniform r-partite hypergraph, the vertex set can be covered by at most t+1 sets, each connected in some color. Our main result is that the conjecture is true for 1 <= t <= r-1. We also prove a slightly weaker result for t >= r, namely that t+2 sets, each connected in some color, are enough to cover the vertex set. To build a bridge between complete r-uniform and complete r-uniform r-partite hypergraphs, we introduce a new notion. A hypergraph is complete r-uniform (r,l)-partite if it has all r-sets that intersect each partite class in at most l vertices. Extending our results achieved for l=1, we prove that for any r >= 3, 2 <= l <= r, k >= 1+r-l, in every spanning k-coloring of the edges of a complete r-uniform (r,l)-partite hypergraph, the vertex set can be covered by at most 1+\lfloor \frac{k-r+\ell-1}{\ell}\rfloor sets, each connected in some color., Comment: 14 pages

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014., Comment: Discrete Mathematics, 2015

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2\alpha(G)$ monochromatic cycles, where $\alpha(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1+\eps){3|V(G)|\over 4}$ can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that $\overline{G}$ does not contain a fixed bipartite graph $H$, we show that in every $2$-edge-coloring of $G$, $|V(G)|-c(H)$ vertices can be covered by two vertex disjoint paths of different colors, where $c(H)$ is a constant depending only on $H$. In particular, we prove that $c(C_4)=1$, which is best possible.

A {\em cross-free} set of size $m$ in a Steiner triple system $(V,{\cal{B}})$ is three pairwise disjoint $m$-element subsets $X_1,X_2,X_3\subset V$ such that no $B\in {\cal{B}}$ intersects all the three $X_i$-s. We conjecture that for every admissible $n$ there is an STS$(n)$ with a cross-free set of size $\lfloor{n-3\over 3}\rfloor$ which if true, is best possible. We prove this conjecture for the case $n=18k+3$, constructing an STS$(18k+3)$ containing a cross-free set of size $6k$. We note that some of the $3$-bichromatic STSs, constructed by Colbourn, Dinitz and Rosa, have cross-free sets of size close to $6k$ (but cannot have size exactly $6k$). The constructed STS$(18k+3)$ shows that equality is possible for $n=18k+3$ in the following result: in every $3$-coloring of the blocks of any Steiner triple system STS$(n)$ there is a monochromatic connected component of size at least $\lceil{2n\over 3}\rceil+1$ (we conjecture that equality holds for every admissible $n$). The analogue problem can be asked for $r$-colorings as well, if $r-1 \equiv 1,3 \mbox{ (mod 6)}$ and $r-1$ is a prime power, we show that the answer is the same as in case of complete graphs: in every $r$-coloring of the blocks of any STS$(n)$, there is a monochromatic connected component with at least ${n\over r-1}$ points, and this is sharp for infinitely many $n$., Comment: Journal of Combinatorial Designs, 2014

We determine the $2$-color Ramsey number of a {\em connected} triangle matching $c(nK_3)$ which is any connected graph containing $n$ vertex disjoint triangles. We obtain that $R(c(nK_3),c(nK_3))=7n-2$, somewhat larger than in the classical result of Burr, Erd\H os and Spencer for a triangle matching, $R(nK_3,nK_3)=5n$. The motivation is to determine the Ramsey number $R(C_n^2,C_n^2)$ of the square of a cycle $C_n^2$. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an "almost" square of a cycle $C_n^{2,c}$ (a cycle of length $n$ in which all but at most a constant number $c$ of short diagonals are present) is asymptotic to $7n/3$., Comment: Journal of Graph Theory, 2015

Motivated by the problem of Gallai on $(1-1)$-transversals of $2$-intervals, it was proved by the authors in 1969 that if the edges of a complete graph $K$ are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced $C_4$ and $C_5$ then the vertices of $K$ can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic $C_4$ and there is no induced $C_5$ in {\em one of the colors}. Here this is strengthened further, it is enough to assume that there is no monochromatic induced $C_4$ and there is no $K_5$ on which both color classes induce a $C_5$. We also answer a question of Kaiser and Rabinovich, giving an example of six $2$-convex sets in the plane such that any three intersect but there is no $(1-1)$-transversal for them.

Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $\chi(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph are colored with $t$ colors in any manner, there is a monochromatic copy of $T$. We observe that $\chi(T,t)$ is well defined and $$\left\lceil {R^r(T,t)-1\over r-1}\right \rceil +1 \le \chi(T,t)\le |E(T)|^t+1$$ where $R^r(T,t)$ is the $t$-color Ramsey number of $H$. We give linear upper bounds for $\chi(T,t)$ when T is a matching or star, proving that for $r\ge 2, k\ge 1, t\ge 1$, $\chi(M_k^r,t)\le (t-1)(k-1)+2k$ and $\chi(S_k^r,t)\le t(k-1)+2$ where $M_k^r$ and $S_k^r$ are, respectively, the $r$-uniform matching and star with $k$ edges. The general bounds are improved for $3$-uniform hypergraphs. We prove that $\chi(M_k^3,2)=2k$, extending a special case of Alon-Frankl-Lov\'asz' theorem. We also prove that $\chi(S_2^3,t)\le t+1$, which is sharp for $t=2,3$. This is a corollary of a more general result. We define $H^{[1]}$ as the 1-intersection graph of $H$, whose vertices represent hyperedges and whose edges represent intersections of hyperedges in exactly one vertex. We prove that $\chi(H)\le \chi(H^{[1]})$ for any $3$-uniform hypergraph $H$ (assuming $\chi(H^{[1]})\ge 2$). The proof uses the list coloring version of Brooks' theorem., Comment: 10 pages

We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$ vertices. The second is the theorem of Cockayne and Lorimer stating that for positive integers satisfying $n_1=\max\{n_1,n_2,\dots,n_t\}$ and with $n=n_1+1+\sum_{i=1}^t (n_i-1)$, the following holds. In every coloring of the edges of $K_n$ with colors $1,2\dots,t$ there is a monochromatic matching of size $n_i$ for some $i\in \{1,2,\dots,t\}$.

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$'s. Several choices of parameters relate to known results and conjectures.

We explore properties of $3$-uniform hypergraphs $H$ without linear cycles. Our main results are that these hypergraphs must contain a vertex of strong degree at most two and must have independent sets of size at least ${2|V(H)|\over 5}$., Comment: 9 pages, 4 figures

A hypergraph is 2-intersecting if any two edges intersect in at least two vertices. Blais, Weinstein and Yoshida asked (as a first step to a more general problem) whether every 2-intersecting hypergraph has a vertex coloring with a constant number of colors so that each hyperedge e has at least min{|e|,3} colors. We show that there is such a coloring with at most 5 colors (which is best possible).

I state some open problems coming from joint work with Paul Erd\H{o}s, Comment: This is a paper form of the talk I gave on July 5, 2013 at the centennial conference in Budapest to honor Paul Erd\H{o}s

A class of graphs closed under taking induced subgraphs is $\chi$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G) \leq f(\omega(G))$. We consider the following question initially studied in [A. Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $\chi$-bounded class $\cal C$, is the class $\bar{C}$ $\chi$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $\chi$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $\chi$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $\chi$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $\chi$-bounded. For every $j$, we construct a class of graphs $\chi$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $\chi$-bounded.

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph K_n with t colors. Another area is to find the minimum number of monochromatic members of F that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for fixed positive integers s,t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of F in every edge coloring of K_n with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: in case of s=t-1 we determine how many vertices can be covered by s monochromatic matchings in every t-coloring of K_n.

Given an $r$-uniform hypergraph $H$, the multicolor Ramsey number $r_k(H)$ is the minimum $n$ such that every $k$-coloring of the edges of the complete $r$-uniform hypergraph $K_n^r$ yields a monochromatic copy of $H$. We investigate $r_k(H)$ when $k$ grows and $H$ is fixed. For nontrivial 3-uniform hypergraphs $H$, the function $r_k(H)$ ranges from $\sqrt{6k}(1+o(1))$ to double exponential in $k$. We observe that $r_k(H)$ is polynomial in $k$ when $H$ is $r$-partite and at least single-exponential in $k$ otherwise. Erd\H{o}s, Hajnal and Rado gave bounds for large cliques $K_s^r$ with $s\ge s_0(r)$, showing its correct exponential tower growth. We give a proof for cliques of all sizes, $s>r$, using a slight modification of the celebrated stepping-up lemma of Erd\H{o}s and Hajnal. For 3-uniform hypergraphs, we give an infinite family with sub-double-exponential upper bound and show connections between graph and hypergraph Ramsey numbers. Specifically, we prove that $$r_k(K_3)\le r_{4k}(K_4^3-e)\le r_{4k}(K_3)+1,$$ where $K_4^3-e$ is obtained from $K_4^3$ by deleting an edge. We provide some other bounds, including single-exponential bounds for $F_5=\{abe,abd,cde\}$ as well as asymptotic or exact values of $r_k(H)$ when $H$ is the bow $\{abc,ade\}$, kite $\{abc,abd\}$, tight path $\{abc,bcd,cde\}$ or the windmill $\{abc,bde,cef,bce\}$. We also determine many new "small" Ramsey numbers and show their relations to designs. For example, the lower bound for $r_6(kite)=8$ is demonstrated by decomposing the triples of $[7]$ into six partial STS (two of them are Fano planes)., Comment: 20 pages, 1 figure

An edge coloring of a tournament $T$ with colors $1,2,\dots,k$ is called \it $k$-transitive \rm if the digraph $T(i)$ defined by the edges of color $i$ is transitively oriented for each $1\le i \le k$. We explore a conjecture of the second author: For each positive integer $k$ there exists a (least) $p(k)$ such that every $k$-transitive tournament has a dominating set of at most $p(k)$ vertices. We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erd\H os, Sands, Sauer and Woodrow (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of B\'ar\'any and Lehel on covering point sets by boxes. The principle used leads also to an upper bound $O(2^{2^{d-1}}d\log d)$ on the $d$-dimensional box-cover number that is better than all previous bounds, in a sense close to best possible. We also improve the best bound known in 3-dimensions from $3^{14}$ to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.

In this paper we consider properly edge-colored graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching. A matching is called \it rainbow \rm if its edges have different colors. The minimum degree of a graph is denoted by $\delta(G)$. We show that properly edge colored graphs $G$ with $|V(G)|\ge 4\delta(G)-3$ have rainbow matchings of size $\delta(G)$, this gives the best known estimate to a recent question of Wang. Since one obviously needs at least $2\delta(G)$ vertices to guarantee a rainbow matching of size $\delta(G)$, we investigate what happens when $|V(G)|\ge 2\delta(G)$. We show that any properly edge colored graph $G$ with $|V(G)|\ge 2\delta$ contains a rainbow matching of size at least $\delta - 2\delta(G)^{2/3}$. This result extends (with a weaker error term) the well-known result that a factorization of the complete bipartite graph $K_{n,n}$ has a rainbow matching of size $n-o(n)$, or equivalently that every Latin square of order $n$ has a partial transversal of size $n-o(n)$ (an asymptotic version of the Ryser - Brualdi conjecture). In this direction we also show that every Latin square of order $n$ has a {\em cycle-free partial transversal} of size $n-o(n)$.

The {\em chromatic gap} is the difference between the chromatic number and the clique number of a graph. Here we investigate $\gap(n)$, the maximum chromatic gap over graphs on $n$ vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey theory and matching theory leads to a simple and (almost) exact formula for $\gap(n)$ in terms of Ramsey numbers., Comment: 41 pages; Cahiers Leibniz No 184, https://consult-cahiersleibniz.g-scop.grenoble-inp.fr/apps/WebObjects/CahiersLeibniz/

Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A_1,...,A_t of independent vertices. A set $U=\cup_{i\in S} A_i$ is called a dominating set of size |S| if for any vertex $v\in \cup_{i\notin S} A_i$ there is a w in U such that (w,v) is in E(D). Let beta(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h=h(beta(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colorings, edge colorings of graphs without 3-colored triangles., Comment: 15 pages, 8 figures