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A hypergraph $H=(V(H), E(H))$ is a Berge copy of a graph $F$, if $V(F)\subset V(H)$ and there is a bijection $f:E(F)\rightarrow E(H)$ such that for any $e\in E(F)$ we have $e\subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain any Berge copies of $F$. We address the saturation problem concerning Berge-$F$-free hypergraphs, i.e., what is the minimum number $sat_r(n,F)$ of hyperedges in an $r$-uniform Berge-$F$-free hypergraph $H$ with the property that adding any new hyperedge to $H$ creates a Berge copy of $F$. We prove that $sat_r(n,F)$ grows linearly in $n$ if $F$ is either complete multipartite or it possesses the following property: if $d_1\le d_2\le \dots \le d_{|V(F)|}$ is the degree sequence of $F$, then $F$ contains two adjacent vertices $u,v$ with $d_F(u)=d_1$, $d_F(v)=d_2$. In particular, the Berge-saturation number of regular graphs grows linearly in $n$., 9 pages

For posets $P$ and $Q$, extremal and saturation problems about weak and strong $P$-free subposets of $Q$ have been studied mostly in the case $Q$ is the Boolean poset $Q_n$, the poset of all subsets of an $n$-element set ordered by inclusion. In this paper, we study some instances of the problem with $Q$ being the grid, and its connections to the Boolean case and to the forbidden submatrix problem.

A Berge-path of length $k$ in a hypergraph $\mathcal{H}$ is a sequence $v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_{i+1}\in e_i,e_{i+1}$ for all $i\in[k]$. F\"uredi, Kostochka and Luo, and independently Gy\H{o}ri, Salia and Zamora determined the maximum number of hyperedges in an $n$-vertex, connected, $r$-uniform hypergraph that does not contain a Berge-path of length $k$ provided $k$ is large enough compared to $r$. They also determined the unique extremal hypergraph $\mathcal{H}_1$. We prove a stability version of this result by presenting another construction $\mathcal{H}_2$ and showing that any $n$-vertex, connected, $r$-uniform hypergraph without a Berge-path of length $k$, that contains more than $|\mathcal{H}_2|$ hyperedges must be a subhypergraph of the extremal hypergraph $\mathcal{H}_1$, provided $k$ is large enough compared to $r$.

In this paper we introduce a unifying approach to the generalized Tur��n problem and supersaturation results in graph theory. The supersaturation-extremal function $satex(n, F : m, G)$ is the least number of copies of a subgraph $G$ an $n$-vertex graph can have, which contains at least $m$ copies of $F$ as a subgraph. We present a survey, discuss previously known results and obtain several new ones focusing mainly on proof methods, extremal structure and phase transition phenomena. Finally we point out some relation with extremal questions concerning hypergraphs, particularly Berge-type results., 16 pages

In this paper we initiate a systematic study of the Tur��n problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the edge-order. A subgraph of an edge-ordered graph is itself an edge-ordered graph with the induced edge-order. We say that an edge-ordered graph $G$ $\textit{avoids}$ another edge-ordered graph $H$, if no subgraph of $G$ is isomorphic to $H$. The $\textit{Tur��n number}$ of an edge-ordered graph $H$ is the maximum number of edges in an edge-ordered graph on $n$ vertices that avoids $H$. We study this problem in general, and establish an Erd��s-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Tur��n number of an edge-ordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Tur��n numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in Discrete Geometry., 41 pages. Updated grants

A subgraph $H$ of $G$ is \textit{singular} if the vertices of $H$ either have the same degree in $G$ or have pairwise distinct degrees in $G$. The largest number of edges of a graph on $n$ vertices that does not contain a singular copy of $H$ is denoted by $T_S(n,H)$. Caro and Tuza [Theory and Applications of Graphs, 6 (2019), 1--32] obtained the asymptotics of $T_S(n,H)$ for every graph $H$, but determined the exact value of this function only in the case $H=K_3$ and $n\equiv 2$ (mod 4). We determine $T_S(n,K_3)$ for all $n\equiv 0$ (mod 4) and $n\equiv 1$ (mod 4), and also $T_S(n,K_{r+1})$ for large enough $n$ that is divisible by $r$. We also explore the connection to the so-called $H$-WORM colorings (colorings without rainbow or monochromatic copies of $H$) and obtain new results regarding the largest number of edges that a graph with an $H$-WORM coloring can have.

As a variant of the famous Tur��n problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of integers $r$ and large enough $n$. For every tree $T$, we determine $\mathrm{rex}(n,T)$ for every $n$ large enough.

Erd\H{o}s and Moser raised the question of determining the maximum number of maximal cliques or equivalently, the maximum number of maximal independent sets in a graph on $n$ vertices. Since then there has been a lot of research along these lines. A $k$-dominating independent set is an independent set $D$ such that every vertex not contained in $D$ has at least $k$ neighbours in $D$. Let $mi_k(n)$ denote the maximum number of $k$-dominating independent sets in a graph on $n$ vertices, and let $\zeta_k:=\lim_{n \rightarrow \infty} \sqrt[n]{mi_k(n)}$. Nagy initiated the study of $mi_k(n)$. In this article we disprove a conjecture of Nagy and prove that for any even $k$ we have $$1.489 \approx \sqrt[9]{36} \le \zeta^k_k.$$ We also prove that for any $k \ge 3$ we have $$\zeta_k^{k} \le 2.053^{\frac{1}{1.053+1/k}}< 1.98,$$ improving the upper bound of Nagy., Comment: 11 pages

Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than $6/7$ of the order of a graph., 21 pages

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free, was recently determined by Alishahi and Taherkhani, whenever $n$ is large enough compared to $k$ and $F$. In this paper, we determine the second largest size of a vertex set $W \subseteq V$ such that $K(n,k)[W]$ is $F$-free, in the case when $F$ is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of $F$. These results generalize the celebrated Erd\H os-Ko-Rado theorem and Hilton-Milner theorem., 12 pages, the proof of Lemma 2.4 is corrected

In this short note we consider the oriented vertex Tur��n problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented hypercube $\overrightarrow{Q_n}$ such that the induced subgraph $\overrightarrow{Q_n}[U]$ does not contain any copy of $\overrightarrow{F}$. We obtain the exact value of $ex_v(\overrightarrow{P_k}, \overrightarrow{Q_n})$ for the directed path $\overrightarrow{P_k}$, the exact value of $ex_v(\overrightarrow{V_2}, \overrightarrow{Q_n})$ for the directed cherry $\overrightarrow{V_2}$ and the asymptotic value of $ex_v(\overrightarrow{T}, \overrightarrow{Q_n})$ for any directed tree $\overrightarrow{T}$.

We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the discrete Pompeiu property. Using this description and the revealed connection we prove that Fuglede's conjecture holds for $\mathbb{Z}_{p^n q^2}$, where $p$ and $q$ are different primes. In particular, we show that every spectral subset of $\mathbb{Z}_{p^n q^2}$ tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede's conjecture holds for $\mathbb{Z}_p^2$., 24 pages. Incorporated referees' and editor's remarks. Corrected typos

In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph $\mathcal{H}$ by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices dominated so far. The game is over if all vertices of $\mathcal{H}$ are dominated. Dominator aims to finish the game as soon as possible, while Staller aims to delay the end of the game. If each player plays optimally and Dominator starts, the length of the game is the invariant `game domination number' denoted by $\gamma_g(\mathcal{H})$. This definition is the generalization of the domination game played on graphs and it is a special case of the transversal game on hypergraphs. After some basic general results, we establish an asymptotically tight upper bound on the game domination number of $k$-uniform hypergraphs. In the remaining part of the paper we prove that $\gamma_g(\mathcal{H}) \le 5n/9$ if $\mathcal{H}$ is a 3-uniform hypergraph of order $n$ and does not contain isolated vertices. This also implies the following new result for graphs: If $G$ is an isolate-free graph on $n$ vertices and each of its edges is contained in a triangle, then $\gamma_g(G) \le 5n/9$.

In combinatorial group testing problems Questioner needs to find a special element $x \in [n]$ by testing subsets of $[n]$. Tapolcai et al. introduced a new model, where each element knows the answer for those queries that contain it and each element should be able to identify the special one. Using classical results of extremal set theory we prove that if $\mathcal{F}_n \subset 2^{[n]}$ solves the non-adaptive version of this problem and has minimal cardinality, then $$\lim_{n \rightarrow \infty} \frac{|\mathcal{F}_n|}{\log_2 n} = \log_{(3/2)}2.$$ This improves results by Tapolcai et al. We also consider related models inspired by secret sharing models, where the elements should share information among them to find out the special one. Finally the adaptive versions of the different models are investigated.

In combinatorial group testing problems Questioner needs to find a defective element $x\in [n]$ by testing subsets of $[n]$. In [18] the authors introduced a new model, where each element knows the answer for those queries that contain it and each element should be able to identify the defective one. In this article we continue to investigate this kind of models with more defective elements. We also consider related models inspired by secret sharing models, where the elements should share information among them to find out the defectives. Finally the adaptive versions of the different models are also investigated.

Given a graph $H$ and a set of graphs $\mathcal F$, let $ex(n,H,\mathcal F)$ denote the maximum possible number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. We investigate the function $ex(n,H,\mathcal F)$, when $H$ and members of $\mathcal F$ are cycles. Let $C_k$ denote the cycle of length $k$ and let $\mathscr C_k=\{C_3,C_4,\ldots,C_k\}$. Some of our main results are the following. (i) We show that $ex(n, C_{2l}, C_{2k}) = ��(n^l)$ for any $l, k \ge 2$. Moreover, we determine it asymptotically in the following cases: We show that $ex(n,C_4,C_{2k}) = (1+o(1)) \frac{(k-1)(k-2)}{4} n^2$ and that the maximum possible number of $C_6$'s in a $C_8$-free bipartite graph is $n^3 + O(n^{5/2})$. (ii) Solymosi and Wong proved that if Erd��s's Girth Conjecture holds, then for any $l \ge 3$ we have $ex(n,C_{2l},\mathscr C_{2l-1})=��(n^{2l/(l-1)})$. We prove that forbidding any other even cycle decreases the number of $C_{2l}$'s significantly: For any $k > l$, we have $ex(n,C_{2l},\mathscr C_{2l-1} \cup \{C_{2k}\})=��(n^2).$ More generally, we show that for any $k > l$ and $m \ge 2$ such that $2k \neq ml$, we have $ex(n,C_{ml},\mathscr C_{2l-1} \cup \{C_{2k}\})=��(n^m).$ (iii) We prove $ex(n,C_{2l+1},\mathscr C_{2l})=��(n^{2+1/l}),$ provided a strong version of Erd��s's Girth Conjecture holds (which is known to be true when $l = 2, 3, 5$). Moreover, forbidding one more cycle decreases the number of $C_{2l+1}$'s significantly: More precisely, we have $ex(n, C_{2l+1}, \mathscr C_{2l} \cup \{C_{2k}\}) = O(n^{2-\frac{1}{l+1}}),$ and $ex(n, C_{2l+1}, \mathscr C_{2l} \cup \{C_{2k+1}\}) = O(n^2)$ for $l > k \ge 2$. (iv) We also study the maximum number of paths of given length in a $C_k$-free graph, and prove asymptotically sharp bounds in some cases., 37 Pages; Substantially revised, contains several new results. Mistakes corrected based on the suggestions of a referee

A classical result of Bondy and Simonovits in extremal graph theory states that if a graph on $n$ vertices contains no cycle of length $2k$ then it has at most $O(n^{1+1/k})$ edges. However, matching lower bounds are only known for $k=2,3,5$. In this paper we study ordered variants of this problem and prove some tight estimates for a certain class of ordered cycles that we call bordered cycles. In particular, we show that the maximum number of edges in an ordered graph avoiding bordered cycles of length at most $2k$ is $��(n^{1+1/k})$. Strengthening the result of Bondy and Simonovits in the case of 6-cycles, we also show that it is enough to forbid these bordered orderings of the 6-cycle to guarantee an upper bound of $O(n^{4/3})$ on the number of edges., 10 pages, 1 figure; added references and some discussion

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph., 18 pages. There was a wrong statement in the first version, it is corrected now

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Tur��n number of Berge-$\mathcal{F}$ is the maximum possible number of edges in an $r$-uniform hypergraph on $n$ vertices containing no Berge-$F$ as a subhypergraph (for every $F \in \mathcal{F}$) and is denoted by $ex_r(n,\mathcal{F})$. We determine the asymptotics for the Tur��n number of Berge-$K_{2,t}$ by showing $$ex_3(n,K_{2,t})=\frac{1}{6}(t-1)^{3/2} \cdot n^{3/2}(1+o(1))$$ for any given $t \ge 7$. We study the analogous question for linear hypergraphs and show that $$ex_3(n,\{C_2, K_{2,t}\}) = \frac{1}{6}\sqrt{t-1} \cdot n^{3/2}(1+o_{t}(1)).$$ We also prove general upper and lower bounds on the Tur��n numbers of a class of graphs including $ex_r(n, K_{2,t})$, $ex_r(n,\{C_2, K_{2,t}\})$, and $ex_r(n, C_{2k})$ for $r \ge 3$. Our bounds improve results of Gerbner and Palmer, F��redi and ��zkahya, Timmons, and provide a new proof of a result of Jiang and Ma., 25 pages; Writing improved based on the suggestions of the referees and mistakes corrected

We study factor of i.i.d. processes on the $d$-regular tree for $d \geq 3$. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where $k$ denotes the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails., 18 pages, 5 figures

We consider the following combinatorial search problem: we are given some excellent elements of $[n]$ and we should find at least one, asking questions of the following type: "Is there an excellent element in $A \subset [n]$?". G.O.H. Katona proved sharp results for the number of questions needed to ask in the adaptive, non-adaptive and two-round versions of this problem. We verify a conjecture of Katona by proving that in the $r$-round version we need to ask $rn^{1/r}+O(1)$ queries for fixed $r$ and this is sharp. We also prove bounds for the queries needed to ask if we want to find at least $d$ excellent elements., 14 pages

We study two models of the Majority problem. We are given n balls and an unknown coloring of them with two colors. We can ask sets of balls of size k as queries, and in the so-called General Model the answer to a query shows if all the balls in the set are of the same color or not. In the so-called Counting Model the answer to a query gives the difference between the cardinalities of the color classes in the query. Our goal is to show a ball of the larger color class, or prove that the color classes are of the same size, using as few queries as possible. In this paper we improve the bounds given by De Marco and Kranakis for the number of queries needed., We cut the non-adaptive results from the first version to publish separately

Suppose we are given a set of $n$ balls $\{b_1,\ldots,b_n\}$ each colored either red or blue in some way unknown to us. To find out some information about the colors, we can query any triple of balls $\{b_{i_1},b_{i_2},b_{i_3}\}$. As an answer to such a query we obtain (the index of) a {\em majority ball}, that is, a ball whose color is the same as the color of another ball from the triple. Our goal is to find a {\em non-minority ball}, that is, a ball whose color occurs at least $\frac n2$ times among the $n$ balls. We show that the minimum number of queries needed to solve this problem is $��(n)$ in the adaptive case and $��(n^3)$ in the non-adaptive case. We also consider some related problems.

Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\mathcal{F} \subset \mathcal{P}([n])$ a \textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \in \mathcal{F}$ with: $$(i) \ \ p|F \setminus G|=q|G \setminus F|, \ \textrm{and}$$ $$(ii) \ f > g \ \textrm{for all} \ f \in F \setminus G \ \textrm{and} \ g \in G \setminus F.$$ Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\sqrt{\log n}}\ \frac{2^n}{\sqrt{n}}).$$ We improve and generalize this result, and prove that the cardinality of every ($p,q$)-tilted Sperner family with patterns on [$n$] is $$O(\sqrt{\log n} \ \frac{2^n}{\sqrt{n}}).$$, 8 pages

There are many generalizations of the Erd��s-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical geometry. We establish the conjecture that for $n \geq t(k-t)+2$ such a family can have at most ${n+k-t-1\choose k-t}$ members.