arXiv subjects

József Balogh

Publications and source records attributed to József Balogh.

At least 127 records · Page 7Linked to original sources

arXiv2015

The typical structure of maximal triangle-free graphs

Recently, settling a question of Erdős, Balogh and Petříčková showed that there are at most $2^{n^2/8+o(n^2)}$ $n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemerédi removal lemma, the Erdős-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest.

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arXiv2015

Intersecting families of discrete structures are typically trivial

The study of intersecting structures is central to extremal combinatorics. A family of permutations $\mathcal{F} \subset S_n$ is \emph{$t$-intersecting} if any two permutations in $\mathcal{F}$ agree on some $t$ indices, and is \emph{trivial} if all permutations in $\mathcal{F}$ agree on the same $t$ indices. A $k$-uniform hypergraph is \emph{$t$-intersecting} if any two of its edges have $t$ vertices in common, and \emph{trivial} if all its edges share the same $t$ vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for $n$ sufficiently large with respect to $t$, the largest $t$-intersecting families in $S_n$ are the trivial ones. The classic Erdős--Ko--Rado theorem shows that the largest $t$-intersecting $k$-uniform hypergraphs are also trivial when $n$ is large. We determine the \emph{typical} structure of $t$-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollobás set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces.

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arXiv2014

Maximum density of an induced 5-cycle is achieved by an iterated blow-up of a 5-cycle

Let $C(n)$ denote the maximum number of induced copies of 5-cycles in graphs on $n$ vertices. For $n$ large enough, we show that $C(n)=a\cdot b\cdot c \cdot d \cdot e + C(a)+C(b)+C(c)+C(d)+C(e)$, where $a+b+c+d+e = n$ and $a,b,c,d,e$ are as equal as possible. Moreover, if $n$ is a power of 5, we show that the unique graph on $n$ vertices maximizing the number of induced 5-cycles is an iterated blow-up of a 5-cycle.

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arXiv2014

On the number of $K_4$-saturating edges

Let $G$ be a $K_4$-free graph, an edge in its complement is a $K_4$-\emph{saturating} edge if the addition of this edge to $G$ creates a copy of $K_4$. Erdős and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with $\lfloor n^2/4\rfloor+1$ edges, one can find at least $(1+o(1))\frac{n^2}{16}$ $K_4$-saturating edges. We construct a graph with only $\frac{2n^2}{33}$ $K_4$-saturating edges. Furthermore, we prove that it is best possible, i.e., one can always find at least $(1+o(1))\frac{2n^2}{33}$ $K_4$-saturating edges in an $n$-vertex $K_4$-free graph with $\lfloor n^2/4\rfloor+1$ edges.

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arXiv2014

Subdivisions of a large clique in $C_6$-free graphs

Mader conjectured that every $C_4$-free graph has a subdivision of a clique of order linear in its average degree. We show that every $C_6$-free graph has such a subdivision of a large clique. We also prove the dense case of Mader's conjecture in a stronger sense, i.e. for every $c$, there is a $c'$ such that every $C_4$-free graph with average degree $cn^{1/2}$ has a subdivision of a clique $K_\ell$ with $\ell=\lfloor c'n^{1/2}\rfloor$ where every edge is subdivided exactly $3$ times.

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arXiv2014

Minimum number of monotone subsequences of length 4 in permutations

We show that for every sufficiently large $n$, the number of monotone subsequences of length four in a permutation on $n$ points is at least $\binom{\lfloor n/3 \rfloor}{4} + \binom{\lfloor(n+1)/3\rfloor}{4} + \binom{\lfloor (n+2)/3\rfloor}{4}$. Furthermore, we characterize all permutations on $[n]$ that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic $K_4$'s is minimized. We show that all the extremal colorings must contain monochromatic $K_4$'s only in one of the two colors. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

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arXiv2014

A random version of Sperner's theorem

Let $\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal{P}(n,p)$ be obtained from $\mathcal{P}(n)$ by selecting elements from $\mathcal{P}(n)$ independently at random with probability $p$. A classical result of Sperner asserts that every antichain in $\mathcal{P}(n)$ has size at most that of the middle layer, $\binom{n}{\lfloor n/2 \rfloor}$. In this note we prove an analogous result for $\mathcal{P} (n,p)$: If $pn \rightarrow \infty$ then, with high probability, the size of the largest antichain in $\mathcal{P}(n,p)$ is at most $(1+o(1)) p \binom{n}{\lfloor n/2 \rfloor}$. This solves a conjecture of Osthus who proved the result in the case when $pn/\log n \rightarrow \infty$. Our condition on $p$ is best-possible. In fact, we prove a more general result giving an upper bound on the size of the largest antichain for a wider range of values of $p$.

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arXiv2014

The number of the maximal triangle-free graphs

Paul Erdős suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among others the Ruzsa-Szemerédi triangle removal lemma, and recent results on characterizing of the structure of independent sets in hypergraphs.

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arXiv2014

The number of maximal sum-free subsets of integers

Cameron and Erdős raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal sum-free sets in $\{1, \dots , n\}$. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, Sós and Temkin on the structure of sum-free sets.

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arXiv2014

The typical structure of sparse $K_{r+1}$-free graphs

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of $H$-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph $H$. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical $H$-free graph with $n$ vertices and $m$ edges changes as $m$ grows from $0$ to $\text{ex}(n,H)$. In this paper, we resolve this problem in the case when $H$ is a clique, extending a classical result of Kolaitis, Prömel, and Rothschild. In particular, we prove that for every $r \ge 2$, there is an explicit constant $θ_r$ such that, letting $m_r = θ_r n^{2-\frac{2}{r+2}} (\log n)^{1/\left[\binom{r+1}{2}-1\right]}$, the following holds for every positive constant $\varepsilon$. If $m \ge (1+\varepsilon) m_r$, then almost all $K_{r+1}$-free $n$-vertex graphs with $m$ edges are $r$-partite, whereas if $n \ll m \le (1-\varepsilon)m_r$, then almost all of them are not $r$-partite.

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arXiv2014

Independent sets in hypergraphs

Many important theorems in combinatorics, such as Szemerédi's theorem on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about independent sets in uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called sparse random setting. This line of research culminated recently in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the well-known conjecture of Kohayakawa, Łuczak and Rödl, a probabilistic embedding lemma for sparse graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, and obtain their natural counting versions, which in some cases are considerably stronger. We moreover prove a sparse version of the Erdős-Frankl-Rödl Theorem on the number of H-free graphs and extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, non-symmetric setting. We remark that similar results have been discovered independently by Saxton and Thomason, and that, in parallel to this work, Conlon, Gowers, Samotij and Schacht have proved a sparse analogue of the counting lemma for subgraphs of the random graph G(n,p), which may be viewed as a version of the KŁR conjecture that is stronger in some ways and weaker in others.

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arXiv2013

On the Chromatic Thresholds of Hypergraphs

Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

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arXiv2013

Hypergraphs with Zero Chromatic Threshold

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erdős-Simonovits. One interesting question, first proposed by Łuczak-Thomassé and then solved by Allen-Böttcher-Griffiths-Kohayakawa-Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turán density that have zero chromatic threshold. In this paper, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a special product of the Bollobás-Erdős graph defined earlier by the authors.

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arXiv2013

Towards a weighted version of the Hajnal-Szemerédi Theorem

For a positive integer r>=2, a K_r-factor of a graph is a collection vertex-disjoint copies of K_r which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemerédi asserts that every graph on n vertices with minimum degree at least (1-1/r)n contains a K_r-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect K_r-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r>=2 and a real t in [0,1] is given. What is the minimum weighted degree of K_n that guarantees the existence of a K_r-factor such that every factor has total edge weight at least tr(r-1)/2? We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as n goes to infinity.

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arXiv2013

On perfect packings in dense graphs

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect H-packings: Given positive integers n, r, D, we characterise the edge density threshold that ensures a perfect K_r-packing in any graph G on n vertices and with minimum degree at least D. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect H-packing. Other related embedding problems are also considered. Indeed, we give a structural result concerning K_r-free graphs that satisfy a certain degree sequence condition.

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arXiv2012

On the tree packing conjecture

The Gyárfás tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},..., T_{n-1}$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$. We show that $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_{n+1}$ (for $n$ large enough). We also prove that any set of $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices pack into $K_{n}$ (for $n$ large enough). Finally, we prove that $t=1/4n^{1/3}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$ as long as each tree has maximum degree at least $2n^{2/3}$ (for $n$ large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Komlós, Sárközy and Szemerédi.

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arXiv2012

Graph bootstrap percolation

Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph $H$, and a set $G \subset E(K_n)$ of initially `infected' edges, we infect, at each time step, a new edge $e$ if there is a copy of $H$ in $K_n$ such that $e$ is the only not-yet infected edge of $H$. We say that $G$ percolates in the $H$-bootstrap process if eventually every edge of $K_n$ is infected. The extremal questions for this model, when $H$ is the complete graph $K_r$, were solved (independently) by Alon, Kalai and Frankl almost thirty years ago. In this paper we study the random questions, and determine the critical probability $p_c(n,K_r)$ for the $K_r$-process up to a poly-logarithmic factor. In the case $r = 4$ we prove a stronger result, and determine the threshold for $p_c(n,K_4)$.

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arXiv2012

Random sum-free subsets of Abelian groups

We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group $G$. In particular, we determine the threshold $p_c \approx \sqrt{\log n / n}$ above which, with high probability as $|G| \to \infty$, each such subset is contained in a maximum-size sum-free subset of $G$, whenever $q$ divides $|G|$ for some (fixed) prime $q$ with $q \equiv 2 \pmod 3$. Moreover, in the special case $G = \ZZ_{2n}$, we determine a sharp threshold for the above property. The proof uses recent 'transference' theorems of Conlon and Gowers, together with stability theorems for sum-free subsets of Abelian groups.

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