arXiv subjects

József Balogh

Publications and source records attributed to József Balogh.

At least 73 records · Page 4Linked to original sources

arXiv2021

Solving Turán's Tetrahedron Problem for the $\ell_2$-Norm

Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron $K_4^3$. This is equivalent to determining the maximum $\ell_1$-norm of the codegree vector of a $K_4^3$-free $n$-vertex $3$-uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, $\text{co}_2(G)$, of a $3$-uniform hypergraph $G$ is the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $xy$, or in other words, the square of the $\ell_2$-norm of the codegree vector of the pairs of vertices. We define $\text{exco}_2(n,H)$ to be the maximum $\text{co}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$. We use flag algebra computations to determine asymptotically the codegree squared extremal number for $K_4^3$ and $K_5^3$ and additionally prove stability results. In particular, we prove that the extremal $K_4^3$-free hypergraphs in $\ell_2$-norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about $\text{exco}_2(n,H)$ including the existence of a scaled limit, blow-up invariance and a supersaturation result.

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arXiv2021

Monochromatic connected matchings in 2-edge-colored multipartite graphs

A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following Łuczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.

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arXiv2021

Chain method for panchromatic colorings of hypergraphs

We deal with an extremal problem concerning panchromatic colorings of hypergraphs. A vertex $r$-coloring of a hypergraph $H$ is \emph{panchromatic} if every edge meets every color. We prove that for every $3<r\leq\sqrt[3]{n/(100\ln n)}$, every $n$-uniform hypergraph $H$ with $|E(H)|\leq \frac{1}{20r^2}\left(\frac{n}{\ln n}\right)^{\frac {r-1}{r}}\left(\frac{r}{r-1}\right)^{n-1}$ has a panchromatic coloring with $r$ colors.

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arXiv2021

A Strengthening of the Erdős-Szekeres Theorem

The Erdős-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone increasing path with $s$ edges. Although $rs + 1$ is the minimum number of vertices needed for this result, not all edges of $K_{rs+1}$ are necessary. We characterize the subgraphs of $K_{rs+1}$ with this coloring property as follows: they are exactly the subgraphs that contain all the edges of a graph we call the circus tent graph $CT(r,s)$. Additionally, we use similar proof techniques to improve upon some of the bounds on the online ordered size Ramsey number of a path given by Pérez-Giménez, Pralat, and West.

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arXiv2021

Generalized rainbow Turán numbers of odd cycles

Given graphs $F$ and $H$, the generalized rainbow Turán number $\text{ex}(n,F,\text{rainbow-}H)$ is the maximum number of copies of $F$ in an $n$-vertex graph with a proper edge-coloring that contains no rainbow copy of $H$. B. Janzer determined the order of magnitude of $\text{ex}(n,C_s,\text{rainbow-}C_t)$ for all $s\geq 4$ and $t\geq 3$, and a recent result of O. Janzer implied that $\text{ex}(n,C_3,\text{rainbow-}C_{2k})=O(n^{1+1/k})$. We prove the corresponding upper bound for the remaining cases, showing that $\text{ex}(n,C_3,\text{rainbow-}C_{2k+1})=O(n^{1+1/k})$. This matches the known lower bound for $k$ even and is conjectured to be tight for $k$ odd.

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arXiv2021

On the number of sum-free triplets of sets

We count the ordered sum-free triplets of subsets in the group $\mathbb{Z}/p\mathbb{Z}$, i.e., the triplets $(A,B,C)$ of sets $A,B,C \subset \mathbb{Z}/p\mathbb{Z}$ for which the equation $a+b=c$ has no solution with $a\in A$, $b \in B$ and $c \in C$. Our main theorem improves on a recent result by Semchankau, Shabanov, and Shkredov using a different and simpler method. Our proof relates previous results on the number of independent sets of regular graphs by Kahn, Perarnau and Perkins, and Csikvári to produce explicit estimates on smaller order terms. We also obtain estimates for the number of sum-free triplets of subsets in a general abelian group.

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arXiv2021

Counting $r$-graphs without forbidden configurations

One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erdős, Kleitman and Rothschild, who showed that the number of $K_r$-free graphs on $n$ vertices is $2^{\text{ex}(n,K_r)+o(n^2)}$. Their work was later extended to forbidding graphs as induced subgraphs by Prömel and Steger. Here, we consider one of the most basic counting problems for $3$-graphs. Let $E_1$ be the $3$-graph with $4$ vertices and $1$ edge. What is the number of induced $\{K_4^3,E_1\}$-free $3$-graphs on $n$ vertices? We show that the number of such $3$-graphs is of order $n^{Θ(n^2)}$. More generally, we determine asymptotically the number of induced $\mathcal{F}$-free $3$-graphs on $n$ vertices for all families $\mathcal{F}$ of $3$-graphs on $4$ vertices. We also provide upper bounds on the number of $r$-graphs on $n$ vertices which do not induce $i \in L$ edges on any set of $k$ vertices, where $L \subseteq \big \{0,1,\ldots,\binom{k}{r} \big\}$ is a list which does not contain $3$ consecutive integers in its complement. Our bounds are best possible up to a constant multiplicative factor in the exponent when $k = r+1$. The main tool behind our proof is counting the solutions of a constraint satisfaction problem.

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arXiv2021

An upper bound on the size of Sidon sets

In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.

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arXiv2021

On stability of the Erdős-Rademacher Problem

Mantel's theorem states that every $n$-vertex graph with $\lfloor \frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the Erdős-Rademacher problem. Lovász and Simonovits proved that there are at least $t\lfloor n/2 \rfloor$ triangles in each of those graphs. Katona and Xiao considered the same problem under the additional condition that there are no $s-1$ vertices covering all triangles. They settled the case $t=1$ and $s=2$. Solving their conjecture, we determine the minimum number of triangles for every fixed pair of $s$ and $t$, when $n$ is sufficiently large. Additionally, solving another conjecture of Katona and Xiao, we extend the theory for considering cliques instead of triangles.

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arXiv2021

Monochromatic paths and cycles in $2$-edge-colored graphs with large minimum degree

A graph $G$ arrows a graph $H$ if in every $2$-edge-coloring of $G$ there exists a monochromatic copy of $H$. Schelp had the idea that if the complete graph $K_n$ arrows a small graph $H$, then every "dense" subgraph of $K_n$ also arrows $H$, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large $n$, if $n = 3t+r$ where $r \in \{0,1,2\}$ and $G$ is an $n$-vertex graph with $δ(G) \ge (3n-1)/4$, then for every $2$-edge-coloring of $G$, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same color, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the same color. Our result is tight in the sense that no longer cycles (of length $>2t+r$) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large $n$, every $(3t-1)$-vertex graph $G$ with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with $2n$ vertices. Moreover, it implies for sufficiently large $n$ the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every $n$-vertex graph $G$ with $δ(G) \ge 3n/4$, in each $2$-edge-coloring of $G$ there exists a monochromatic cycle of length at least $2t+r$.

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arXiv2021

Max Cuts in Triangle-free Graphs

A well-known conjecture by Erdős states that every triangle-free graph on $n$ vertices can be made bipartite by removing at most $n^2/25$ edges. This conjecture was known for graphs with edge density at least $0.4$ and edge density at most $0.172$. Here, we will extend the edge density for which this conjecture is true; we prove the conjecture for graphs with edge density at most $0.2486$ and for graphs with edge density at least $0.3197$. Further, we prove that every triangle-free graph can be made bipartite by removing at most $n^2/23.5$ edges improving the previously best bound of $n^2/18$.

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arXiv2021

Maximum size intersecting families of bounded minimum positive co-degree

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $δ_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $δ_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\H os-Ko-Rado theorem which corresponds to the case $k=1$. Our proof is based on the delta-system method.

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arXiv2021

Maximum Number of Almost Similar Triangles in the Plane

A triangle $T'$ is $\varepsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\varepsilon$. Given a triangle $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, Bárány and Füredi asked to determine the maximum number of triangles $h(n,T,\varepsilon)$ being $\varepsilon$-similar to $T$ in a planar point set of size $n$. We show that for almost all triangles $T$ there exists $\varepsilon=\varepsilon(T)>0$ such that $h(n,T,\varepsilon)=n^3/24 (1+o(1))$. Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof.

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arXiv2020

An efficient container lemma

We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.

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arXiv2020

Maximum determinant and permanent of sparse 0-1 matrices

We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles.

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arXiv2020

Ramsey upper density of infinite graph factors

The study of upper density problems on Ramsey theory was initiated by Erdős and Galvin in 1993. In this paper we are concerned with the following problem: given a fixed finite graph $F$, what is the largest value of $λ$ such that every 2-edge-coloring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite $F$-factor whose vertex set has upper density at least $λ$? Here we prove a new lower bound for this problem. For some choices of $F$, including cliques and odd cycles, this new bound is sharp, as it matches an older upper bound. For the particular case where $F$ is a triangle, we also give an explicit lower bound of $1-\frac{1}{\sqrt{7}}=0.62203\dots$, improving the previous best bound of 3/5.

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arXiv2020

Independent sets in the middle two layers of Boolean lattice

For an odd integer $n=2d-1$, let $\mathcal{B}(n, d)$ be the subgraph of the hypercube $Q_n$ induced by the two largest layers. In this paper, we describe the typical structure of independent sets in $\mathcal{B}(n, d)$ and give precise asymptotics on the number of them. The proofs use Sapozhenko's graph container method and a recently developed method of Jenssen and Perkins, which combines Sapozhenko's graph container lemma with the cluster expansion for polymer models from statistical physics.

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arXiv2020

Long rainbow arithmetic progressions

Define $T_k$ as the minimal $t\in \mathbb{N}$ for which there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[tn]$ for all $n\in \mathbb{N}$. Jungić, Licht (Fox), Mahdian, Nesetril and Radoicić proved that $\lfloor{\frac{k^2}{4}\rfloor}\le T_k$. We almost close the gap between the upper and lower bounds by proving that $T_k \le k^2e^{(\ln\ln k)^2(1+o(1))}$. Conlon, Fox and Sudakov have independently shown a stronger statement that $T_k=O(k^2\log k)$.

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