arXiv subjects

Dániel Gerbner

Publications and source records attributed to Dániel Gerbner.

At least 73 records · Page 4Linked to original sources

arXiv2021

On non-adaptive majority problems of large query size

We are given $n$ balls and an unknown coloring of them with two colors. Our goal is to find a ball that belongs to the larger color class, or show that the color classes have the same size. We can ask sets of $k$ balls as queries, and the problem has different variants, according to what the answers to the queries can be. These questions has attracted several researchers, but the focus of most research was the adaptive version, where queries are decided sequentially, after learning the answer to the previous query. Here we study the non-adaptive version, where all the queries have to be asked at the same time.

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arXiv2021

Forbidden subposet problems in the grid

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.

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arXiv2021

Turán problems for Edge-ordered graphs

In this paper we initiate a systematic study of the Turán 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 $\textit{Turán number}$ 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 Erdős-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Turán number of an edge-ordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Turán 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.

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arXiv2021

A note on the uniformity threshold for Berge hypergraphs

A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Grósz, Methuku and Tompkins in 2020 showed that for any graph $F$, there is an integer $r_0=r_0(F)$, such that for any $r\ge r_0$, any $r$-uniform hypergraph without a Berge copy of $F$ has $o(n^2)$ hyperedges. The smallest such $r_0$ is called the uniformity threshold of $F$ and is denoted by $th(F)$. They showed that $th(F)\le R(F,F')$, where $R$ denotes the off-diagonal Ramsey number and $F'$ is any graph obtained form $F$ by deleting an edge. We improve this bound to $th(F)\le R(K_{χ(F)},F')$, and use the new bound to determine $th(F)$ exactly for several classes of graphs.

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arXiv2021

Unified approach to the generalized Turán problem and supersaturation

In this paper we introduce a unifying approach to the generalized Turán 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.

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arXiv2021

A non-aligning variant of generalized Turán problems

In the so-called generalized Turán problems we study the largest number of copies of $H$ in an $n$-vertex $F$-free graph $G$. Here we introduce a variant, where $F$ is not forbidden, but we restrict how copies of $H$ and $F$ can be placed in $G$. More precisely, given an integer $n$ and graphs $H$ and $F$, what is the largest number of copies of $H$ in an $n$-vertex graph such that the vertex set of that copy does not contain and is not contained in the vertex set of a copy of $F$? We solve this problem for some instances, give bounds in other instances, and we use our results to determine the generalized Turán number for some pairs of graphs.

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arXiv2021

Generalized Turán problems for $K_{2,t}$

We study the generalized Turán function $ex(n,H,F)$, when $H$ or $F$ is $K_{2,t}$. We determine the order of magnitude of $ex(n,H,K_{2,t})$ when $H$ is a tree, and determine its asymptotics for a large class of trees. We also determine the asymptotics of $ex(n,K_{2,t},H)$ in most cases.

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arXiv2021

A note on the Turán number of a Berge odd cycle

In this note we obtain upper bounds on the number of hyperedges in 3-uniform hypergraphs not containing a Berge cycle of given odd length. We improve the bound given by Füredi and Özkahya in 2017. The result follows from a more general theorem. We also obtain some new results for Berge cliques.

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arXiv2021

Generalized Turán problems for complete bipartite graphs

For graph $G$, $F$ and integer $n$, the generalized Tuán number $ex(n,G,F)$ denotes the maximum number of copies of $G$ that an $F$-free $n$-vertex graph can have. We study this parameter when both $G$ and $F$ are complete bipartite graphs.

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arXiv2021

On saturation of Berge hypergraphs

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 $n$.

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arXiv2021

A note on stability for maximal $F$-free graphs

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently extended to odd cycles in place of $K_3$ by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.

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arXiv2020

On Turán-good graphs

For graphs $H$ and $F$, the generalized Turán number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We say that $H$ is $F$-Turán-good if $ex(n,H,F)$ is the number of copies in the $(χ(F)-1)$-partite Turán graph, provided $n$ is large enough. We present a general theorem in case $F$ has an edge whose deletion decreases the chromatic number. In particular, this determines $ex(n,P_k,C_{2\ell+1})$ and $ex(n,C_{2k},C_{2\ell+1})$ exactly, if $n$ is large enough. We also study the case when $F$ has a vertex whose deletion decreases the chromatic number.

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arXiv2020

Saturation problems with regularity constraints

For a graph $F$, we say that another graph $G$ is $F$-saturated, if $G$ is $F$-free and adding any edge to $G$ would create a copy of $F$. We study for a given graph $F$ and integer $n$ whether there exists a regular $n$-vertex $F$-saturated graph, and if it does, what is the smallest number of edges of such a graph. We mainly focus on the case when $F$ is a complete graph and prove for example that there exists a $K_3$-saturated regular graph on $n$ vertices for every large enough $n$. We also study two relaxed versions of the problem: when we only require that no regular $F$-free supergraph of $G$ should exist or when we drop the $F$-free condition and only require that any newly added edge should create a new copy of $F$.

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arXiv2020

Rainbow Ramsey problems for the Boolean lattice

We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We consider both weak and strong (non-induced and induced) versions of this problem. We also investigate related problems on (partial) $k$-colorings of $B_n$ that do not admit rainbow antichains of size $k$.

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arXiv2020

Supersaturation, counting, and randomness in forbidden subposet problems

In the area of forbidden subposet problems we look for the largest possible size $La(n,P)$ of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain a forbidden inclusion pattern described by $P$. The main conjecture of the area states that for any finite poset $P$ there exists an integer $e(P)$ such that $La(n,P)=(e(P)+o(1))\binom{n}{\lfloor n/2\rfloor}$. In this paper, we formulate three strengthenings of this conjecture and prove them for some specific classes of posets. (The parameters $x(P)$ and $d(P)$ are defined in the paper.) $\bullet$ For any finite connected poset $P$ and $\varepsilon>0$, there exists $δ>0$ and an integer $x(P)$ such that for any $n$ large enough, and $\mathcal{F}\subseteq 2^{[n]}$ of size $(e(P)+\varepsilon)\binom{n}{\lfloor n/2\rfloor}$, $\mathcal{F}$ contains at least $δn^{x(P)}\binom{n}{\lfloor n/2\rfloor}$ copies of $P$. $\bullet$ The number of $P$-free families in $2^{[n]}$ is $2^{(e(P)+o(1))\binom{n}{\lfloor n/2\rfloor}}$. $\bullet$ For any finite poset $P$, there exists a positive rational $d(P)$ such that if $p=ω(n^{-d(P)})$, then the size of the largest $P$-free family in $\mathcal{P}(n,p)$ is $(e(P)+o(1))p\binom{n}{\lfloor n/2\rfloor}$ with high probability.

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arXiv2020

Combinatorics in the exterior algebra and the Bollobás Two Families Theorem

We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the underlying vector space onto subspaces, and the interior product, we find analogues of local and global LYM inequalities, the Erdős-Ko-Rado theorem, and the Ahlswede-Khachatrian bound for $t$-intersecting hypergraphs. Using these tools, we prove a new extension of the Two Families Theorem of Bollobás, giving a weighted bound for subspace configurations satisfying a skew cross-intersection condition. We also verify a recent conjecture of Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins, and Xiao on pairs of set systems satisfying both an intersection and a cross-intersection condition.

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arXiv2020

Generalized Turán problems for small graphs

For graphs $H$ and $F$, the generalized Turán number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We consider this problem when both $H$ and $F$ have at most four vertices. We give sharp results in almost all cases, and connect the remaining cases to well-known unsolved problems. Our main new contribution is applying the progressive induction method of Simonovits for generalized Turán problems.

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arXiv2020

Some exact results for generalized Turán problems

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Turán graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough). We say that such a graph $H$ is $F$-Turán-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph $H$, there is $k$ large enough such that $H$ is $K_k$-Turán-good. (ii) The path $P_3$ is $F$-Turán-good for $F$ with $χ(F) \geq 4$. (iii) The path $P_4$ and cycle $C_4$ are $C_5$-Turán-good. (iv) The cycle $C_4$ is $F_2$-Turán-good where $F_2$ is the graph of two triangles sharing exactly one vertex.

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