arXiv subjects

Dániel Gerbner

Publications and source records attributed to Dániel Gerbner.

At least 55 records · Page 3Linked to original sources

arXiv2023

On the number of $A$-transversals in hypergraphs

A set $S$ of vertices in a hypergraph is \textit{strongly independent} if every hyperedge shares at most one vertex with $S$. We prove a sharp result for the number of maximal strongly independent sets in a $3$-uniform hypergraph analogous to the Moon-Moser theorem. Given an $r$-uniform hypergraph ${\mathcal H}$ and a non-empty set $A$ of non-negative integers, we say that a set $S$ is an \textit{$A$-transversal} of ${\mathcal H}$ if for any hyperedge $H$ of ${\mathcal H}$, we have \mbox{$|H\cap S| \in A$}. Independent sets are $\{0,1,\dots,r{-}1\}$-transversals, while strongly independent sets are $\{0,1\}$-transversals. Note that for some sets $A$, there may exist hypergraphs without any $A$-transversals. We study the maximum number of $A$-transversals for every $A$, but we focus on the more natural sets, e.g., $A=\{a\}$, $A=\{0,1,\dots,a\}$ or $A$ being the set of odd or the set of even numbers.

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arXiv2023

Cooperation in Combinatorial Search

In the game theoretical approach of the basic problem in Combinatorial Search an adversary thinks of a defective element $d$ of an $n$-element pool $X$, and the questioner needs to find $x$ by asking questions of type is $d\in Q$? for certain subsets $Q$ of $X$. We study cooperative versions of this problem, where there are multiple questioners, but not all of them learn the answer to the queries. We consider various models that differ in how it is decided who gets to ask the next query, who obtains the answer to the query, and who needs to know the defective element by the end of the process.

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arXiv2023

Generalized Turán results for disjoint cliques

The generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We denote by $tF$ the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of $\mathrm{ex}(n,K_s,tK_r)$. We extend this result in three directions. First, we determine $\mathrm{ex}(n,K_s,tK_r)$ exactly for sufficiently large $n$. Second, we determine the asymptotics of the analogous number for $p$-uniform hypergraphs. Third, we determine the order of magnitude of $\mathrm{ex}(n,H,tK_r)$ for every graph $H$, and also of the analogous number for $p$-uniform hypergraphs.

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arXiv2023

A note on strongly and totally chain intersecting families

Bernáth and Gerbner in 2007 introduced $(p,q)$-chain intersecting families of subsets of an $n$-element underlying set. Those have the property that for any $p$-chain $A_1\subsetneq A_2\subsetneq \dots \subsetneq A_p$ and $q$-chain $B_1\subsetneq B_2\subsetneq \dots \subsetneq B_q$, we have $A_p\cap B_q\neq \emptyset$. Bernáth and Gerbner determined the largest cardinality of such families. They also introduced strongly $(p,q)$-chain intersecting families, where $A_p\cap B_1\neq \emptyset$ and totally $(p,q)$-chain intersecting families, where $A_1\cap B_1\neq \emptyset$. They obtained some partial results on the maximum cardinality of such families. We extend those results by determining the largest cardinality of strongly $(p,q)$-chain intersecting families if $n$ is sufficiently large, and by determining the largest cardinality of totally $(2,2)$-chain intersecting families.

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arXiv2023

On non-degenerate Berge-Turán problems

Given a hypergraph $\mathcal{H}$ and a graph $G$, we say that $\mathcal{H}$ is a \textit{Berge}-$G$ if there is a bijection between the hyperedges of $\mathcal{H}$ and the edges of $G$ such that each hyperedge contains its image. We denote by $ex_k(n,\text{Berge-}F)$ the largest number of hyperedges in a $k$-uniform Berge-$F$-free graph. Let $ex(n,H,F)$ denote the largest number of copies of $H$ in $n$-vertex $F$-free graphs. It is known that $ex(n,K_k,F)\le ex_k(n,\text{Berge-}F)\le ex(n,K_k,F)+ex(n,F)$, thus if $χ(F)>r$, then $ex_k(n,\text{Berge-}F)=(1+o(1)) ex(n,K_k,F)$. We conjecture that $ex_k(n,\text{Berge-}F)=ex(n,K_k,F)$ in this case. We prove this conjecture in several instances, including the cases $k=3$ and $k=4$. We prove the general bound $ex_k(n,\text{Berge-}F)= ex(n,K_k,F)+O(1)$.

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arXiv2022

On weakly Turán-good graphs

Given graphs $H$ and $F$ with $χ(H)<χ(F)$, we say that $H$ is weakly $F$-Turán-good if among $n$-vertex $F$-free graphs, a $(χ(F)-1)$-partite graph contains the most copies of $H$. Let $H$ be a bipartite graph that contains a complete bipartite subgraph $K$ such that each vertex of $H$ is adjacent to a vertex of $K$. We show that $H$ is weakly $K_3$-Turán-good, improving a very recent asymptotic bound due to Grzesik, Gy\H ori, Salia and Tompkins. They also showed that for any $r$ there exist graphs that are not weakly $K_r$-Turán-good. We show that for any non-bipartite $F$ there exists graphs that are not weakly $F$-Turán-good. We also show examples of graphs that are $C_{2k+1}$-Turán-good but not $C_{2\ell+1}$-Turán-good for every $k>\ell$.

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arXiv2022

Rainbow copies of $F$ in families of $H$

We study the following problem. How many distinct copies of $H$ can an $n$-vertex graph $G$ have, if $G$ does not contain a rainbow $F$, that is, a copy of $F$ where each edge is contained in a different copy of $H$? The case $H=K_r$ is equivalent to the Turán problem for Berge hypergraphs, which has attracted several researchers recently. We also explore the connection of our problem to the so-called generalized Turán problems. We obtain several exact results. In the particularly interesting symmetric case where $H=F$, we completely solve the case $F$ is the 3-edge path, and asymptitically solve the case $F$ is a book graph.

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arXiv2022

On graphs that contain exactly k copies of a subgraph, and a related problem in search theory

We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Turán number $\mathrm{ex}(n,F)$ that is among the most studied parameters in extremal graph theory. We show that for any $F$ and $k$, $\mathrm{exa}_k(n,F)=(1+o(1))\mathrm{ex}(n,F))$ and determine the exact values of $\mathrm{exa}_k(n,K_3)$ and $\mathrm{exa}_1(n,K_r)$ for $n$ large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order $n$ that consists of an unknown copy of $F$ and some isolated vertices. We can ask pairs of vertices as queries, and the answer tells us whether there is an edge between those vertices. Our goal is to describe the graph using as few queries as possible. Aigner and Triesch in 1990 showed that the number of queries needed is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$. Among other results we show that the number of queries that were answered NO is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$.

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arXiv2022

On the extremal graphs in generalized Turán problems

Given two graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in an $n$-vertex $F$-free graph. For every $F$ and sufficiently large $n$, we present an extremal graph for a generalized Turán problem, i.e., an $F$-free $n$ vertex graph $G$ that for some $H$ contains exactly $\mathrm{ex}(n,H,F)$ copies of $H$.

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arXiv2022

Some exact results for non-degenerate generalized Turán problems

The generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $χ(H)<χ(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the extremal graph is a complete $(χ(F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs.

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arXiv2022

The maximum number of triangles in $F_k$-free graphs

The generalized Turán number $ex(n,K_s,H)$ is the maximum number of complete graph $K_s$ in an $H$-free graph on $n$ vertices. Let $F_k$ be the friendship graph consisting of $k$ triangles. Erdős and Sós (1976) determined the value of $ex(n,K_3,F_2)$. Alon and Shikhelman (2016) proved that $ex(n,K_3, F_k)\le (9k-15)(k+1)n.$ In this paper, by using a method developed by Chung and Frankl in hypergraph theory, we determine the exact value of $ex(n,K_3,F_k)$ and the extremal graph for any $F_k$ when $n\ge 4k^3$.

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arXiv2022

Paths are Turán-good

We show that among $K_{k+1}$-free $n$-vertex graphs, the Turán graph contains the most copies of any path.

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arXiv2022

Some stability and exact results in generalized Turán problems

Given graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains almost $\mathrm{ex}(n,H,F)$ copies of $H$, than $G$ is in some sense similar to some extremal graph. We obtain new stability results for generalized Turán problems and derive several new exact results.

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arXiv2022

A note on the number of triangles in graphs without the suspension of a path on four vertices

The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex connected to each of the four vertices, and is denoted by $\hat{P_4}$. The largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted by $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $ ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$, $ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.

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arXiv2022

The profile polytope of non-trivial intersecting families

The profile vector of a family $\mathcal{F}$ of subsets of an $n$-element set is $(f_0,f_1, \ldots, f_n)$ where $f_i$ denotes the number of the $i$-element members of $\mathcal{F}$. In this paper we determine the extreme points of the set of profile vectors for the class of non-trivial intersecting families.

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arXiv2022

Generalized Turán results for intersecting cliques

For fixed graphs $F$ and $H$, the generalized Turán problem asks for the maximum number $ex(n,H,F)$ of copies of $H$ that an $n$-vertex $F$-free graph can have. In this paper, we focus on cases with $F$ being $B_{r,s}$, the graph consisting of two cliques of size $s$ sharing $r$ common vertices. We determine $ex(n,K_t,B_{r,0})$, $ex(n,K_t,B_{r,1})$ and $ex(n,K_{a,b},B_{3,1})$ for all values of $a,b,r,t$ if $n$ is large enough.

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arXiv2022

Generalized Turán problems for double stars

We study the generalized Turán function $ex(n,H,F)$, when $H$ or $F$ is a double star $S_{a,b}$, which is a tree with a central edge $uv$, $a$ leaves connected to $u$ and $b$ leaves connected to $v$. We determine $ex(n,K_k,S_{a,b})$ and $ex(n,S_{a,b},F)$ for sufficiently large $n$, where $F$ is either a 3-chromatic graph with an edge whose deletion results in a bipartite graph, or the 2-fan, i.e. two triangles sharing a vertex. We also give bounds on $ex(n,S_{a,b},S_{c,d})$.

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arXiv2021

The Turán number of Berge book hypergraphs

Given a graph $G$, a Berge copy of $G$ is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H ori in 2006 showed that for $r=3$ or $r=4$, an $r$-uniform $n$-vertex Berge triangle-free hypergraph has at most $\lfloor n^2/8(r-2)\rfloor$ hyperedges if $n$ is large enough, and this bound is sharp. The book graph $B_t$ consists of $t$ triangles sharing an edge. Very recently, Ghosh, Győri, Nagy-György, Paulos, Xiao and Zamora showed that a 3-uniform $n$-vertex Berge $B_t$-free hypergraph has at most $n^2/8+o(n^2)$ hyperedges if $n$ is large enough. They conjectured that this bound can be improved to $\lfloor n^2/8\rfloor$. We prove this conjecture for $t=2$ and disprove it for $t>2$ by proving the sharp bound $\lfloor n^2/8\rfloor+(t-1)^2$. We also consider larger uniformity and determine the largest number of Berge $B_t$-free $r$-uniform hypergraphs besides an additive term $o(n^2)$. We obtain a similar bound if the Berge $t$-fan ($t$ triangles sharing a vertex) is forbidden.

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