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We prove that suitably generic pairs of linear equations on an even number of variables are uncommon. This verifies a conjecture of Kam\v{c}ev, Morrison and the second author. Moreover, we prove that any large system containing such a $(2\times k)$-system as a minimal subsystem is uncommon., Comment: 11 pages

We address a problem which is a generalization of Tur\'an-type problems recently introduced by Imolay, Karl, Nagy and V\'ali. Let $F$ be a fixed graph and let $G$ be the union of $k$ edge-disjoint copies of $F$, namely $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$, where each $F_i$ is isomorphic to a fixed graph $F$ and $E(F_i)\cap E(F_j)=\emptyset$ for all $i \neq j$. We call a subgraph $H\subseteq G$ multicolored if $H$ and $F_i$ share at most one edge for all $i$. Define $\text{ex}_F(H,n)$ to be the maximum value $k$ such that there exists $G = \mathbin{\dot{\cup}}_{i=1}^{k} F_i$ on $n$ vertices without a multicolored copy of $H$. We show that $\text{ex}_{C_5}(C_3,n) \le n^2/25 + 3n/25+o(n)$ and that all extremal graphs are close to a blow-up of the 5-cycle. This bound is tight up to the linear error term., Comment: 17 pages

Given a family $\mathcal{H}$ of graphs, a graph $G$ is called $\mathcal{H}$-universal if $G$ contains every graph of $\mathcal{H}$ as a subgraph. Following the extensive research on universal graphs of small size for bounded-degree graphs, Alon asked what is the minimum number of edges that a graph must have to be universal for the class of all $n$-vertex graphs that are $D$-degenerate. In this paper, we answer this question up to a factor that is polylogarithmic in $n.$, Comment: 17 pages

A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs $(H_1,\ldots,H_q)$, denoted by $G\to_q(H_1,\ldots,H_q)$, if every $q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$ for some $i\in[q]$. Let $s_q(H_1,\ldots,H_q)$ denote the smallest minimum degree of $G$ over all graphs $G$ that are minimal $q$-Ramsey for $(H_1,\ldots,H_q)$ (with respect to subgraph inclusion). The study of this parameter was initiated in 1976 by Burr, Erd\H{o}s and Lov\'asz, who determined its value precisely for a pair of cliques. Over the past two decades the parameter $s_q$ has been studied by several groups of authors, the main focus being on the symmetric case, where $H_i\cong H$ for all $i\in [q]$. The asymmetric case, in contrast, has received much less attention. In this paper, we make progress in this direction, studying asymmetric tuples consisting of cliques, cycles and trees. We determine $s_2(H_1,H_2)$ when $(H_1,H_2)$ is a pair of one clique and one tree, a pair of one clique and one cycle, and when it is a pair of two different cycles. We also generalize our results to multiple colors and obtain bounds on $s_q(C_\ell,\ldots,C_\ell,K_t,\ldots,K_t)$ in terms of the size of the cliques $t$, the number of cycles, and the number of cliques. Our bounds are tight up to logarithmic factors when two of the three parameters are fixed.

A system of linear forms $L=\{L_1,\ldots,L_m\}$ over $\mathbb{F}_q$ is said to be Sidorenko if the number of solutions to $L=0$ in any $A \subseteq \mathbb{F}_{q}^n$ is asymptotically as $n\to\infty$ at least the expected number of solutions in a random set of the same density. Work of Saad and Wolf (2017) and of Fox, Pham and Zhao (2019) fully characterises single equations with this property and both sets of authors ask about a characterisation of Sidorenko systems of equations. In this paper, we make progress towards this goal. Firstly, we find a simple necessary condition for a system to be Sidorenko, thus providing a rich family of non-Sidorenko systems. In the opposite direction, we find a large family of structured Sidorenko systems, by utilising the entropy method. We also make significant progress towards a full classification of systems of two equations., Comment: 18 pages

A system of linear equations $L$ over $\mathbb{F}_q$ is common if the number of monochromatic solutions to $L$ in any two-colouring of $\mathbb{F}_q^n$ is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of $\mathbb{F}_q^n$. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Building upon earlier work of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have recently been fully characterised by Fox, Pham and Zhao, who asked about common \emph{systems} of equations. In this paper we move towards a classification of common systems of two or more linear equations. In particular we prove that any system containing an arithmetic progression of length four is uncommon, confirming a conjecture of Saad and Wolf. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems., Comment: 20 pages, accepted version

We say that $G \to (F,H)$ if, in every edge colouring $c: E(G) \to \{1,2\}$, we can find either a $1$-coloured copy of $F$ or a $2$-coloured copy of $H$. The well-known Kohayakawa--Kreuter conjecture states that the threshold for the property $G(n,p) \to (F,H)$ is equal to $n^{-1/m_{2}(F,H)}$, where $m_{2}(F,H)$ is given by \[ m_{2}(F,H):= \max \left\{\dfrac{e(J)}{v(J)-2+1/m_2(H)} : J \subseteq F, e(J)\ge 1 \right\}. \] In this paper, we show the $0$-statement of the Kohayakawa--Kreuter conjecture for every pair of cycles and cliques., Comment: 21 pages

We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform hypergraphs on $n$ vertices provided that $dn\le c\binom{n}{k}$ for a constant $c>0$, and $3 \leq k < n^C$ for any $C<1/9.$ Our results relate the degree sequence of a random $k$-uniform hypergraph to a simple model of nearly independent binomial random variables, thus extending the recent results for graphs due to the second and third author.

We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and close the gaps between the results known for the sparse and dense ranges. In the case of bipartite graphs, these results were proved by Greenhill, McKay and Wang in 2006 and by Canfield, Greenhill and McKay in 2008, respectively. Our method also essentially covers the sparse range, for which much less was known in the case of loopless digraphs. For the range of densities which our results cover, they imply that the degree sequence of a random bipartite graph with m edges is accurately modelled by a sequence of independent binomial random variables, conditional upon the sum of variables in each part being equal to m. A similar model also holds for loopless digraphs.

Let $r \ge 4$ be an integer and consider the following game on the complete graph $K_n$ for $n \in r \mathbb{Z}$: Two players, Maker and Breaker, alternately claim previously unclaimed edges of $K_n$ such that in each turn Maker claims one and Breaker claims $b \in \mathbb{N}$ edges. Maker wins if her graph contains a $K_r$-factor, that is a collection of $n/r$ vertex-disjoint copies of $K_r$, and Breaker wins otherwise. In other words, we consider a $b$-biased $K_r$-factor Maker-Breaker game. We show that the threshold bias for this game is of order $n^{2/(r+2)}$. This makes a step towards determining the threshold bias for making bounded-degree spanning graphs and extends a result of Allen et al.\ who resolved the case $r \in \{3,4\}$ up to a logarithmic factor.

Consider a matrix $M$ chosen uniformly at random from a class of $m \times n$ matrices of zeros and ones with prescribed row and column sums. A partially filled matrix $D$ is a $\mathit{defining}$ $\mathit{set}$ for $M$ if $M$ is the unique member of its class that contains the entries in $D$. The $\mathit{size}$ of a defining set is the number of filled entries. A $\mathit{critical}$ $\mathit{set}$ is a defining set for which the removal of any entry stops it being a defining set. For some small fixed $\epsilon>0$, we assume that $n\le m=o(n^{1+\epsilon})$, and that $\lambda\le1/2$, where $\lambda$ is the proportion of entries of $M$ that equal $1$. We also assume that the row sums of $M$ do not vary by more than $\mathcal{O}(n^{1/2+\epsilon})$, and that the column sums do not vary by more than $\mathcal{O}(m^{1/2+\epsilon})$. Under these assumptions we show that $M$ almost surely has no defining set of size less than $\lambda mn-\mathcal{O}(m^{7/4+\epsilon})$. It follows that $M$ almost surely has no critical set of size more than $(1-\lambda)mn+\mathcal{O}(m^{7/4+\epsilon})$. Our results generalise a theorem of Cavenagh and Ramadurai, who examined the case when $\lambda=1/2$ and $n=m=2^k$ for an integer $k$.

In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: For every $c>1/2$ and $\varepsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n\geq n_0$ vertices contains $(1-\varepsilon)cn$ edge-disjoint Hamilton cycles.

A graph $G$ is Ramsey for a graph $H$ if every 2-colouring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a `sparse' graph $G$ that is Ramsey for $H$? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of $H$ are bounded, then there is a graph $G$ with $O(|V(H)|)$ edges that is Ramsey for $H$. This was previously only known for the smaller class of graphs $H$ with bounded bandwidth. On the other hand, we prove that the treewidth of a graph $G$ that is Ramsey for $H$ cannot be bounded in terms of the treewidth of $H$ alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and $H$ is a tree., Comment: Following the release of our original preprint, a positive answer to our Question 5.2 has been given in arXiv:1907.03466 . Also an alternative proof of Theorem 1.3 has been provided by Jonathan Rollin, which we added to the preprint. Subsequently concluding remarks have been changed, the remaining changes are minor

Let $T$ be a tree such that all its vertices of degree more than two lie on one path, that is, $T$ is a caterpillar subdivision. We prove that there exists $\epsilon>0$ such that for every graph $G$ with $|V(G)|\ge 2$ not containing $T$ as an induced subgraph, either some vertex has at least $\epsilon|V(G)|$ neighbours, or there are two disjoint sets of vertices $A,B$, both of cardinality at least $\epsilon|V(G)|$, where there is no edge joining $A$ and $B$. A consequence is: for every caterpillar subdivision $T$, there exists $c>0$ such that for every graph $G$ containing neither of $T$ and its complement as an induced subgraph, $G$ has a clique or stable set with at least $|V(G)|^c$ vertices. This extends a theorem of Bousquet, Lagoutte and Thomass\'e [JCTB 2015], who proved the same when $T$ is a path, and a recent theorem of Choromanski, Falik, Liebenau, Patel and Pilipczuk [Electron. J. Combin. 2018], who proved it when $T$ is a "hook".

For an integer $q\ge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property then $G$ is called $q$-Ramsey-minimal for $H$. Generalising a statement by Burr, Ne\v{s}et\v{r}il and R\"odl from 1977 we prove that, for $q\ge 3$, if $G$ is a graph that is not $q$-Ramsey for some graph $H$ then $G$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$, as long as $H$ is $3$-connected or isomorphic to the triangle. For such $H$, the following are some consequences. (1) For $2\le r< q$, every $r$-Ramsey-minimal graph for $H$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$. (2) For every $q\ge 3$, there are $q$-Ramsey-minimal graphs for $H$ of arbitrarily large maximum degree, genus, and chromatic number. (3) The collection $\{{\cal M}_q(H) : H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where ${\cal M}_q(H)$ denotes the set of all graphs that are $q$-Ramsey-minimal for $H$. We also address the question which pairs of graphs satisfy ${\cal M}_q(H_1)={\cal M}_q(H_2)$, in which case $H_1$ and $H_2$ are called $q$-equivalent. We show that two graphs $H_1$ and $H_2$ are $q$-equivalent for even $q$ if they are $2$-equivalent, and that in general $q$-equivalence for some $q\ge 3$ does not necessarily imply $2$-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Ne\v{s}et\v{r}il and R\"odl and by Fox, Grinshpun, Liebenau, Person and Szab\'o imply that the complete graph is not $2$-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

The celebrated Erd\H{o}s-Hajnal conjecture states that for every proper hereditary graph class $\mathcal{G}$ there exists a constant $\varepsilon = \varepsilon(\mathcal{G}) > 0$ such that every graph $G \in \mathcal{G}$ contains a clique or an independent set of size $|V(G)|^\varepsilon$. Recently, there has been a growing interest in the symmetrized variant of this conjecture, where one additionally requires $\mathcal{G}$ to be closed under complementation. We show that any hereditary graph class that is closed under complementation and excludes a fixed caterpillar as an induced subgraph satisfies the Erd\H{o}s-Hajnal conjecture. Here, a caterpillar is a tree whose vertices of degree at least three lie on a single path (i.e., our caterpillars may have arbitrarily long legs). In fact, we prove a stronger property of such graph classes, called in the literature the strong Erd\H{o}s-Hajnal property: for every such graph class $\mathcal{G}$, there exists a constant $\delta = \delta(\mathcal{G}) > 0$ such that every graph $G \in \mathcal{G}$ contains two disjoint sets $A,B \subseteq V(G)$ of size at least $\delta|V(G)|$ each so that either all edges between $A$ and $B$ are present in $G$, or none of them. This result significantly extends the family of graph classes for which we know that the strong Erd\H{o}s-Hajnal property holds; for graph classes excluding a graph $H$ and its complement it was previously known only for paths [Bousquet, Lagoutte, Thomass\'{e}, JCTB 2015] and hooks (i.e., paths with an additional pendant vertex at third vertex of the path) [Choromanski, Falik, Liebenau, Patel, Pilipczuk, arXiv:1508.00634].

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of $d$-regular graphs for all $d$, as $n\to\infty$., Comment: The new version contains a modification of the main proof that makes the calculation component easier

We prove that, for $n\geq 4$, the graphs $K_n$ and $K_n+K_{n-1}$ are Ramsey equivalent. That is, if $G$ is such that any red-blue colouring of its edges creates a monochromatic $K_n$ then it must also possess a monochromatic $K_n+K_{n-1}$.

The celebrated Erdos-Hajnal conjecture states that for every $n$-vertex undirected graph $H$ there exists $\eps(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or an independent set of size at least $n^{\eps(H)}$. A weaker version of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both $H$ and its complement $H^{\compl}$. We show that the weaker conjecture holds if $H$ is any path with a pendant edge at its third vertex; thus we give a new infinite family of graphs for which the conjecture holds.

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.

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini-Schramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branch-depth assumption nor the representability assumption can be removed., Comment: Accepted to the European Journal of Combinatorics

In the Oriented-cycle game, introduced by Bollob\'as and Szab\'o, two players, called OMaker and OBreaker, alternately direct edges of $K_n$. OMaker directs exactly one edge, whereas OBreaker is allowed to direct between one and $b$ edges. OMaker wins if the final tournament contains a directed cycle, otherwise OBreaker wins. Bollob\'as and Szab\'o conjectured that for a bias as large as $n-3$ OMaker has a winning strategy if OBreaker must take exactly $b$ edges in each round. It was shown recently by Ben-Eliezer, Krivelevich and Sudakov, that OMaker has a winning strategy for this game whenever $b\leq \frac{n}{2}-2$. In this paper, we show that OBreaker has a winning strategy whenever $b\geq \frac{5n}{6}+2$. Moreover, in case OBreaker is required to direct exactly $b$ edges in each move, we show that OBreaker wins for $b\geq \frac{19n}{20}$, provided that $n$ is large enough. This refutes the conjecture by Bollob\'as and Szab\'o., Comment: 31 pages, 7 figures

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.

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph on n vertices and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament T_k on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of T_k; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2-o(1))log_2 n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2-o(1))log_2 n Breaker can prevent that the underlying graph of Maker's graph contains a k-clique. Moreover the precise value of our lower bound differs from the upper bound only by an additive constant of 12. We also discuss the question whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two "clever" players and the game played by two "random" players, is supported by the tournament game: It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid. Finally, we consider the orientation-game version of the tournament game, where Maker wins the game if the final digraph -- containing also the edges directed by Breaker -- possesses a copy of T_k. We prove that in that game Breaker has a winning strategy for k = (4+o(1))log_2 n.

For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n+1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n-1 moves and provide non-trivial examples of families of trees which Maker cannot build in n-1 moves.

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.

In this paper we analyze classical Maker-Breaker games played on the edge set of a sparse random board $G\sim \gnp$. We consider the Hamiltonicity game, the perfect matching game and the $k$-connectivity game. We prove that for $p(n)\geq \text{polylog}(n)/n$, the board $G\sim \gnp$ is typically such that Maker can win these games asymptotically as fast as possible, i.e. within $n+o(n)$, $n/2+o(n)$ and $kn/2+o(n)$ moves respectively.