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We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and $|A + A| \leq m$ is at most \[2^{o(s)}\binom{\frac{m+\beta}{2}}{s},\] where $\beta$ is the size of the largest subgroup of $G$ of size at most $\left(1+o(1)\right)m$. This bound is sharp for $\mathbb{Z}$ and many other groups. Our result improves the one of Campos and nearly bridges the remaining gap in a conjecture of Alon, Balogh, Morris, and Samotij. We also explore the behaviour of uniformly chosen random sets $A \subseteq \{1,\ldots,n\}$ with $|A| = s$ and $|A + A| \leq m$. Under the same assumption that $m \ll s^2/(\log n)^2$, we show that with high probability there exists an arithmetic progression $P \subseteq \mathbb{Z}$ of size at most $m/2 + o(m)$ containing all but $o(s)$ elements of $A$. Analogous results are obtained for asymmetric sumsets, improving results by Campos, Coulson, Serra, and W\"otzel. The main tool behind our proofs is a graph container theorem combined with a variant of an asymmetric hypergraph container theorem., Comment: 36 pages (including appendix)

Consider the Erd\H{o}s-R\'enyi random graph process $\{G_m\}_{m \ge 0}$ in which we start with an empty graph $G_0$ on the vertex set $[n]$, and in each step form $G_i$ from $G_{i-1}$ by adding one new edge chosen uniformly at random. Resolving a conjecture by Benjamini and Tzalik, we give a simple proof that w.h.p. as soon as $G_m$ has minimum degree $2$ it is globally rigid in the following sense: For any function $d \colon E(G_m) \rightarrow \mathbb{R}$, there exists at most one injective function $f \colon [n] \rightarrow \mathbb{R}$ (up to isometry) such that $d(ij) = |f(i) - f(j)|$ for every $ij \in E(G_m)$. We also resolve a related question of Gir\~{a}o, Illingworth, Michel, Powierski, and Scott in the sparse regime for the random graph and give some open problems., Comment: 5 pages

Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is defined to be the sequence of graphs $G_i$, $i\geq 0$ which starts with $G_0:=G$ and in which $G_{i+1}$ is obtained from $G_i$ by adding every edge that completes a copy of $H$. We are interested in the maximum number of steps, over all $n$-vertex graphs $G$, that this process takes to stabilise. In the first of a series of papers exploring the behaviour of this function, denoted $M_H(n)$, and its dependence on certain properties of $H$, we investigate the case when $H$ is a cycle. We determine the running time precisely, giving the first infinite family of graphs $H$ for which an exact solution is known. The maximum running time of the $C_k$-bootstrap process is of the order $\log_{k-1}(n)$ for all $3\leq k\in \mathbb{N}$. Interestingly though, the function exhibits different behaviour depending on the parity of $k$ and the exact location of the values of $n$ for which $M_H(n)$ increases is determined by the Frobenius number of a certain numerical semigroup depending on $k$., Comment: 25 pages, 1 figure

We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general. We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.

We study two related problems concerning the number of homogeneous subsets of given size in graphs that go back to questions of Erd\H{o}s. Most notably, we improve the upper bounds on the Ramsey multiplicity of $K_4$ and $K_5$ and settle the minimum number of independent sets of size $4$ in graphs with clique number at most $4$. Motivated by the elusiveness of the symmetric Ramsey multiplicity problem, we also introduce an off-diagonal variant and obtain tight results when counting monochromatic $K_4$ or $K_5$ in only one of the colors and triangles in the other. The extremal constructions for each problem turn out to be blow-ups of a graph of constant size and were found through search heuristics. They are complemented by lower bounds established using flag algebras, resulting in a fully computer-assisted approach. For some of our theorems we can also derive that the extremal construction is stable in a very strong sense. More broadly, these problems lead us to the study of the region of possible pairs of clique and independent set densities that can be realized as the limit of some sequence of graphs., Comment: 39 pages, 3 figures, published in Foundations of Computational Mathematics (2024), DOI: 10.1007/s10208-024-09675-6

In the max-min allocation problem a set $P$ of players are to be allocated disjoint subsets of a set $R$ of indivisible resources, such that the minimum utility among all players is maximized. We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources. Bez\'akov\'a and Dani showed that this problem is NP-hard to approximate within a factor less than $2$, consequently a great deal of work has focused on approximate solutions. The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been much interest in bounding the integrality gap of this CLP. The existing algorithms and integrality gap estimations are all based one way or another on the combinatorial augmenting tree argument of Haxell for finding perfect matchings in certain hypergraphs. Our main innovation in this paper is to introduce the use of topological methods for the restricted max-min allocation problem, to replace the combinatorial argument. This approach yields substantial improvements in the integrality gap of the CLP. In particular we improve the previously best known bound of $3.808$ to $3.534$. We also study the $(1,\varepsilon)$-restricted version, in which resources can take only two values, and improve the integrality gap in most cases.

In 1985, Mader conjectured that for every acyclic digraph $F$ there exists $K=K(F)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of $F$. This conjecture remains widely open, even for digraphs $F$ on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomass\'{e} studied special cases of Mader's problem and made the following conjecture: for every $\ell \geq 2$ there exists $K = K(\ell)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of every orientation of a cycle of length $\ell$. We prove this conjecture and answer further open questions raised by Aboulker et al., Comment: 28 pages, 3 figures

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vec{\chi}}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $\text{mader}_{\vec{\chi}}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomass\'{e}. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is that $\text{mader}_{\vec{\chi}}(F)=4$ for every tournament $F$ of order $4$. This is an extension of the classical result by Dirac that $4$-chromatic graphs contain a $K_4$-subdivision to directed graphs., Comment: 24 pages, 1 figure

A well-known conjecture, often attributed to Ryser, states that the cover number of an $r$-partite $r$-uniform hypergraph is at most $r - 1$ times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Kir\'{a}ly and T\'{o}thm\'{e}r\'{e}sz considered the problem under the assumption that the hypergraph is $t$-intersecting, conjecturing that the cover number $\tau(\mathcal{H})$ of such a hypergraph $\mathcal{H}$ is at most $r - t$. In these papers, it was proven that the conjecture is true for $r \leq 4t-1$, but also that it need not be sharp; when $r = 5$ and $t = 2$, one has $\tau(\mathcal{H}) \leq 2$. We extend these results in two directions. First, for all $t \geq 2$ and $r \leq 3t-1$, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy $\tau(\mathcal{H}) \leq \lfloor(r - t)/2 \rfloor + 1$. Second, we extend the range of $t$ for which the conjecture is known to be true, showing that it holds for all $r \leq \frac{36}{7}t-5$. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for $k$-wise $t$-intersecting hypergraphs, for all $k \geq 3$ and $t \geq 1$. We further give bounds on the cover numbers of strictly $t$-intersecting hypergraphs and the $s$-cover numbers of $t$-intersecting hypergraphs., Comment: Revised according to the referee reports, updated references, to appear in JCTA

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority 4-coloring and conjectured that every digraph admits a majority 3-coloring. We verify this conjecture for digraphs with chromatic number at most 6 or dichromatic number at most 3. We obtain analogous results for list coloring: We show that every digraph with list chromatic number at most 6 or list dichromatic number at most 3 is majority 3-choosable. We deduce that digraphs with maximum out-degree at most 4 or maximum degree at most 7 are majority 3-choosable. On the way to these results we investigate digraphs admitting a majority 2-coloring. We show that every digraph without odd directed cycles is majority 2-choosable. We answer an open question posed by Kreutzer et al. negatively, by showing that deciding whether a given digraph is majority 2-colorable is NP-complete. Finally we deal with a fractional relaxation of majority coloring proposed by Kreutzer et al. and show that every digraph has a fractional majority 3.9602-coloring. We show that every digraph with minimum out-degree $\Omega\left((1/\varepsilon)^2\ln(1/\varepsilon)\right)$ has a fractional majority $(2+\varepsilon)$-coloring., Comment: 15 pages, 2 figures

Two $n \times n$ Latin squares $L_1, L_2$ are said to be orthogonal if, for every ordered pair $(x,y)$ of symbols, there are coordinates $(i,j)$ such that $L_1(i,j) = x$ and $L_2(i,j) = y$. A $k$-MOLS is a sequence of $k$ pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed $k$, log-asymptotically tight bounds on the number of $k$-MOLS. To study the situation when $k$ grows with $n$, we bound the number of ways a $k$-MOLS can be extended to a $(k+1)$-MOLS. These bounds are again tight for constant $k$, and allow us to deduce upper bounds on the total number of $k$-MOLS for all $k$. These bounds are close to tight even for $k$ linear in $n$, and readily generalize to the broader class of gerechte designs, which include Sudoku squares., Comment: 18 pages

We demonstrate a close connection between the classic planar Singer difference sets and certain norm equation systems arising from projective norm graphs. This, on the one hand leads to a novel description of planar Singer difference sets as a subset $\mathcal{H}$ of $\mathcal{N}$, the group of elements of norm 1 in the field extension $\mathbb{F}_{q^3}/\mathbb{F}_q$. $\mathcal{H}$ is given as the solution set of a simple polynomial equation, and we obtain an explicit formula expressing each non-identity element of $\mathcal{N}$ as a product $B\cdot C^{-1}$ with $B, C\in \mathcal{H}$. The description and the definitions naturally carry over to the nonplanar and the infinite setting. On the other hand, relying heavily on the difference set properties, we also complete the proof that the projective norm graph $\text{NG}(q,4)$ does contain the complete bipartite graph $K_{4,6}$ for every prime power $q \geq 5$. This complements the property, known for more than two decades, that projective norm graphs do not contain $K_{4,7}$ (and hence provide tight lower bounds for the Tur\'an number $ex(n,K_{4,7})$)., Comment: 25 pages + 1 page Appendix

The projective norm graphs $\text{NG}(q,t)$ provide tight constructions for the Tur\'an number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$. In this paper we determine their automorphism group and explore their small subgraphs. To this end we give quite precise estimates on the number of solutions of certain equation systems involving norms over finite fields. The determination of the largest integer $s_t$, such that the projective norm graph $\text{NG}(q,t)$ contains $K_{t,s_t}$ for all large enough prime powers $q$ is an important open question with far-reaching general consequences. The best known bounds, $t-1\leq s_t \leq (t-1)!$, are far apart for $t\geq 4$. Here we prove that $\text{NG}(q,4)$ does contain (many) $K_{4,6}$ for any prime power $q$ not divisble by $2$ or $3$. This greatly extends recent work of Grosu, using a completely different approach. Along the way we also count the copies of any fixed $3$-degenerate subgraph, and find that projective norm graphs are quasirandom with respect to this parameter. Some of these results also extend the work of Alon and Shikhelman on generalized Tur\'an numbers. Finally we also give a new, more elementary proof for the $K_{4,7}$-freeness of $\text{NG}(q,4)$., Comment: 41 pages + 6 pages of Appendix

Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollob\'as in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E_t\subseteq E(K_n)$, we infect a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E_t\cup e)$. A question raised by Bollob\'as asks for the maximum time the process can run before it stabilizes. Bollob\'as, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H=K_r$. They answered the question for $r\leq 4$ and gave a non-trivial lower bound for every $r\geq 5$. They also conjectured that the maximal running time is $o(n^2)$ for every integer $r$. In this paper we disprove their conjecture for every $r\geq 6$ and we give a better lower bound for the case $r=5$; in the proof we use the Behrend construction., Comment: 10 pages

We study the positional game where two players, Maker and Breaker, alternately select respectively $1$ and $b$ previously unclaimed edges of $K_n$. Maker wins if she succeeds in claiming all edges of some odd cycle in $K_n$ and Breaker wins otherwise. Improving on a result of Bednarska and Pikhurko, we show that Maker wins the odd cycle game if $b \leq ((4 - \sqrt{6})/5 + o(1)) n$. We furthermore introduce "connected rules" and study the odd cycle game under them, both in the Maker-Breaker as well as in the Client-Waiter variant., Comment: 22 pages, 3 figures

The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph ${\cal H}$ the main questions is to determine the smallest bias $q({\cal H})$ that allows Breaker to force that Maker ends up with an independent set of ${\cal H}$. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the $H$-building games, studied for graphs by Bednarska and {\L}uczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging., Comment: 26 pages

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. $r$-partite hypergraphs whose cover number is $r-1$ times its matching number. Aside from a few sporadic examples, the set of uniformities $r$ for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order $r-1$ exists. We produce a new infinite family of $r$-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order $r-2$ exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the {\em Ryser poset} of extremal intersecting $r$-partite $r$-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in $\sqrt{r}$. This provides further evidence for the difficulty of Ryser's Conjecture.

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this work we focus on the case of Breaker playing randomly and Maker being "clever". The reverse scenario is treated in a separate paper. We determine the sharp threshold bias of classical games played on the edge set of the complete graph $K_n$, such as connectivity, perfect matching, Hamiltonicity, and minimum degree-$1$. In all of these games, the threshold is equal to the trivial upper bound implied by the number of edges needed for Maker to occupy a winning set. Moreover, we show that the clever Maker can not only win against an asymptotically optimal bias, but can do so very fast, wasting only logarithmically many moves (while the winning set sizes are linear in $n$)., Comment: This is the second part of our work on this subject, the first part can be found here: arXiv:1507.06688 . Originally, both parts were contained in the same paper. This original version can be found here: arXiv:1507.06688v2

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-$k$. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly more tilted towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph., Comment: This paper has been split in two parts, the second part can be found at http://arxiv.org/abs/1602.04628 . A full version from before the split is submission v2 of this arXiv entry, available at http://arxiv.org/abs/1507.06688v2

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) = r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r.

We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced subgraphs of minimum degree $3$. Erd\H{o}s, Faudree, Gy\'arf\'as, and Schelp conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$ for some function $C(n)$ tending to infinity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with $n$ vertices and $2n-2$ edges, containing no proper subgraph of minimum degree $3$., Comment: 22 pages, 14 figures

Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply K\"onig's Theorem and every bipartite graph is a witness for its tightness. The conjecture has also been proven for $r = 3$ by Aharoni using topological methods, but the proof does not give information on the extremal $3$-uniform hypergraphs. Our goal in this paper is to characterize those hypergraphs which are tight for Aharoni's Theorem. Our proof of this characterization is also based on topological machinery, particularly utilizing results on the (topological) connectedness of the independence complex of the line graph of the link graphs of $3$-uniform Ryser-extremal hypergraphs, developed in a separate paper. The current paper contains the second, structural hypergraph-theoretic part of the argument, where we use the information on the line graph of the link graphs to nail down the elements of a structure we call \emph{home-base hypergraph}. While there is a single minimal home-base hypergraph with matching number $k$ for every positive integer $k \in \mathbb{N}$, home-base hypergraphs with matching number $k$ are far from being unique. There are infinitely many of them and each of them is composed of $k$ copies of two different kinds of basic structures, whose hyperedges can intersect in various restricted, but intricate ways. Our characterization also proves an old and wide open strengthening of Ryser's Conjecture, due to Lov\'asz, for the $3$-uniform extremal case, that is, for hypergraphs with $\tau = 2 \nu$.

In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the lower bound $\frac{\dim(\mathcal{I}(G))}{2} - 2$ on the connectedness of the independence complex $\mathcal{I}(G)$ of line graphs of bipartite graphs $G$ is tight. In our main theorem we characterize the extremal examples. Our proof of this characterization is based on topological machinery. Our motivation for studying this problem comes from a classical conjecture of Ryser. Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $(r - 1)$-times the size of the largest matching. For $r = 2$, the conjecture is simply K\"onig's Theorem. It has also been proven for $r = 3$ by Aharoni using a beautiful topological argument. In a separate paper we characterize the extremal examples for the $3$-uniform case of Ryser's Conjecture (i.e., Aharoni's Theorem), and in particular resolve an old conjecture of Lov\'asz for the case of Ryser-extremal $3$-graphs. Our main result in this paper will provide us with valuable structural information for that characterization. Its proof is based on the observation that link graphs of Ryser-extremal $3$-uniform hypergraphs are exactly the bipartite graphs we study here.

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H' are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H'. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph K_k. A famous theorem of Nesetril and Rodl implies that any graph H which is Ramsey-equivalent to K_k must contain K_k. We prove that the only connected graph which is Ramsey-equivalent to K_k is itself. This gives a negative answer to the question of Szabo, Zumstein, and Zurcher on whether K_k is Ramsey-equivalent to K_k.K_2, the graph on k+1 vertices consisting of K_k with a pendent edge. In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph H if it is Ramsey for H but no proper subgraph of G is Ramsey for H. Let s(H) be the smallest minimum degree over all Ramsey minimal graphs for H. The study of s(H) was introduced by Burr, Erdos, and Lovasz, where they show that s(K_k)=(k-1)^2. We prove that s(K_k.K_2)=k-1, and hence K_k and K_k.K_2 are not Ramsey-equivalent. We also address the question of which non-connected graphs are Ramsey-equivalent to K_k. Let f(k,t) be the maximum f such that the graph H=K_k+fK_t, consisting of K_k and f disjoint copies of K_t, is Ramsey-equivalent to K_k. Szabo, Zumstein, and Zurcher gave a lower bound on f(k,t). We prove an upper bound on f(k,t) which is roughly within a factor 2 of the lower bound.

Let r >= s >= 0 be integers and G be an r-graph. The higher inclusion matrix M_s^r(G) is a {0,1}-matrix with rows indexed by the edges of G and columns indexed by the subsets of V(G) of size s: the entry corresponding to an edge e and a subset S is 1 if S is contained in e and 0 otherwise. Following a question of Frankl and Tokushige and a result of Keevash, we define the rank-extremal function rex(n,t,r,s) as the maximum number of edges of an r-graph G having rank M_s^r(G) <=\binom{n}{s} - t. For t at most linear in n we determine this function as well as the extremal r-graphs. The special case t=1 answers a question of Keevash., Comment: 18 pages

Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an r-partite r-uniform multi-hypergraph whose edges are colored with f colors such that every color class is a matching of size t. This problem was posed by Aharoni and Berger, who asked to determine the minimum number of colors which guarantees a rainbow matching. We improve on the known upper bounds for this problem for all values of the parameters. In particular for every fixed r, we give an upper bound which is polynomial in t, improving the superexponential estimate of Alon. Our proof also works in the setting not requiring the hypergraph to be r-partite., Comment: 11 pages

In this paper we study the behaviour of the domination number of the Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. Extending a result of Wieland and Godbole we show that the domination number of $\mathcal{G}(n,p)$ is equal to one of two values asymptotically almost surely whenever $p \gg \frac{\ln^2n}{\sqrt{n}}$. The explicit values are exactly at the first moment threshold, that is where the expected number of dominating sets starts to tend to infinity. For small $p$ we also provide various non-concentration results which indicate why some sort of lower bound on the probability $p$ is necessary in our first theorem. Concentration, though not on a constant length interval, is proven for every $p\gg 1/n$. These results show that unlike in the case of $p \gg \frac{\ln^2n}{\sqrt{n}}$ where concentration of the domination number happens around the first moment threshold, for $p = O(\ln n/n)$ it does so around the median. In particular, in this range the two are far apart from each other.

We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3., Comment: 12 pages

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree $\Delta$ satisfies some basic expansion properties and contains a family of $(1-o(1))\Delta/2$ edge disjoint Hamilton cycles, then there also exists a covering of its edges by $(1+o(1))\Delta/2$ Hamilton cycles. This implies that for every $\alpha >0$ and every $p \geq n^{\alpha-1}$ there exists a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles asymptotically almost surely, which is nearly optimal., Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some math/0612751

The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our paper is to identify several of these problems, among them a couple of widely studied extremal functions related to certain restricted versions of the k-SAT problem, where the Local Lemma does give essentially optimal answers. As our main contribution, we construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e + o(1))*2^k/k clauses. The Lopsided Local Lemma shows that this is asymptotically best possible. The determination of this extremal function is particularly important as it represents the value where the corresponding k-SAT problem exhibits a complexity hardness jump: from having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. The construction of our unsatisfiable CNF-formulas is based on the binary tree approach of [16] and thus the constructed formulas are in the class MU(1) of minimal unsatisfiable formulas having one more clauses than variables. The main novelty of our approach here comes in setting up an appropriate continuous approximation of the problem. This leads us to a differential equation, the solution of which we are able to estimate. The asymptotically optimal binary trees are then obtained through a discretization of this solution. The importance of the binary trees constructed is also underlined by their appearance in many other scenarios. In particular, they give asymptotically precise answers for seemingly unrelated problems like the European Tenure Game introduced by Doerr [9] and a search problem allowing a limited number of consecutive lies., Comment: 40 pages, 1 figure

In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\em who} wins but rather {\em how fast} can one win. These type of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph $K_n$ on $n$ vertices. For several games that are known to be an Enforcer's win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game.

In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of ``small'' sets, the other ensuring the existence of an edge between any two disjoint ``large'' sets. We also discuss applications in positional games, random graphs and extremal graph theory., Comment: 19 pages

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on the edges of random graph $G(n,p)$, for various target families $F$ of winning sets. More generally, for each probability above this threshold we study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game.