Proof of the Pach-Tardos conjecture
We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.
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arXiv subjects
Lior Gishboliner
Publications and source records attributed to Lior Gishboliner.
NP-Hardness of the $H$-Free Edge-Deletion Problem
For a graph $H$, the $H$-freeness edge-deletion problem is the algorithmic problem of finding, for an input graph $G$, the minimum number of edges of $G$ whose deletion turns $G$ into an $H$-free graph. We show that for every graph $H$ containing a cycle, this problem is NP-hard. This proves a conjecture of Gishboliner, Levanzov and Shapira, and completes the characterization of the complexity of the $H$-freeness edge-deletion problem, answering a question of Alon, Shapira and Sudakov.
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Homomorphism and VC-dimension thresholds: spectra and separations
Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic graphs, both the homomorphism and VC-dimension thresholds have infinite spectra and are nonmonotone under taking induced subgraphs. For complete tripartite graphs with a singleton part, we prove $δ_{\mathrm{hom}}(K_{1,s,t}) \ge \max\left\{\frac13,\frac{s}{1+s+t}\right\}$, with equality for an infinite range of $s,t$; in particular, $δ_{\mathrm{hom}}(K_{1,s,s})=s/(2s+1)$ for every $s\ge2$. More generally, for every $r\ge3$, the value $(r-2)/(r-1)$ is an accumulation point of the homomorphism thresholds of $r$-chromatic graphs. For maximal $H$-free graphs, we determine the VC-dimension threshold of every complete tripartite graph and prove that it is positive for every nonbipartite $H$, yielding in particular the exact value for every odd cycle. We also classify the chromatic threshold under an a priori VC-dimension bound. Together with known blowup-threshold results, our theorems reveal that $δ_χ,δ_{\mathrm{hom}},δ_{\mathrm{VC}}$, and $δ_{\mathrm B}$ are \emph{pairwise distinct}: bounded colorability, homomorphic compressibility, neighborhood complexity, and exact blowup structure are genuinely different forms of global simplicity. The proofs develop random and grid-based obstructions to bounded homomorphic images, saturated gadgets that preserve high VC-dimension under maximal completion, and a core-orientation method for raising minimum degree while preserving $H$-freeness.
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Two Erdos-Hajnal-type theorems for forbidden order-size pairs
The celebrated Erdős-Hajnal conjecture says that any graph without a fixed induced subgraph $H$ contains a very large homogeneous set. A direct analog of this conjecture is not true for hypergraphs. In this paper we present two natural variants of this problem which do hold for hypergraphs. We show that for every $r \geq 3$, $m \geq m_0(r)$ and $0 \leq f \leq \binom{m}{r}$, if an $r$-graph $G$ does not contain $m$ vertices spanning exactly $f$ edges, then $G$ contains much larger homogeneous sets than what is guaranteed to exist in general $r$-graphs. We also prove that if a $3$-graph $G$ does not contain homogeneous sets of polynomial size, then for every $m \geq 3$ there are $Ω(m^3)$ values of $f$ such that $G$ contains $m$ vertices spanning exactly $f$ edges. This makes progress on a problem of Axenovich, Bradač, Gishboliner, Mubayi and Weber.
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Asymmetric results about graph homomorphisms
Many important results in extremal graph theory can be roughly summarised as "if a triangle-free graph $G$ has certain properties, then it has a homomorphism to a triangle-free graph $Γ$ of bounded size". For example, bounds on homomorphism thresholds give such a statement if $G$ has sufficiently high minimum degree, and the approximate homomorphism theorem gives such a statement for all $G$, if one weakens the notion of homomorphism appropriately. In this paper, we study asymmetric versions of these results, where the assumptions on $G$ and $Γ$ need not match. For example, we prove that if $G$ is a graph with odd girth at least $9$ and minimum degree at least $δ|G|$, then $G$ is homomorphic to a triangle-free graph whose size depends only on $δ$. Moreover, the odd girth assumption can be weakened to odd girth at least $7$ if $G$ has bounded VC dimension or bounded domination number. This gives a new and improved proof of a result of Huang et al. We also prove that in the asymmetric approximate homomorphism theorem, the bounds exhibit a rather surprising ``double phase transition'': the bounds are super-exponential if $G$ is only assumed to be triangle-free, they become exponential if $G$ is assumed to have odd girth $7$ or $9$, and become linear if $G$ has odd girth at least $11$. Our proofs use a wide variety of techniques, including entropy arguments, the Frieze--Kannan weak regularity lemma, properties of the generalised Mycielskian construction, and recent work on abundance and the asymmetric removal lemma.
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A Simple Counting Argument for Dense Linear Hypergraphs
In connection to the Brown-Erdős-Sós conjecture, we give a short local averaging proof of a density theorem for linear uniform hypergraphs. Let $r \ge 3$, $k \ge 3$, and suppose that $n \ge (r-2)(k-2)+1$. If $H$ is a linear $r$-uniform hypergraph on $n$ vertices and \[|E(H)| \geq \frac{k-2}{r^2((r-2)(k-2)+1)}n^2 + \frac{n}{r},\] then $H$ contains $k$ edges spanning at most $(r-2)k+3$ vertices. In the standard linear-density normalization, this gives the asymptotic density threshold $c \geq \frac{r-1}{r} \cdot \frac{k-2}{(r-2)(k-2)+1} + o(1)$. In particular, this yields a simple proof of the large-uniformity form of the Brown-Erdős-Sós theorem, due to Keevash and Long. In the case of triple systems, our bound becomes $c \geq \frac{2(k-2)}{3(k-1)} + o(1)$, improving upon a bound of $\frac{4}{5}$ due to Santos and Tyomkyn.
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Multicolor $K_r$-Tilings with High Discrepancy
We study the minimum degree threshold $δ_{r,q}$ guaranteeing the existence of $K_r$-tilings of high discrepancy in any $q$-edge-coloring. Balogh, Csaba, Pluhár and Treglown handled the 2-color case, proving that $δ_{r,2} = \frac{r}{r+1}$ for all $r \geq 3$. Here we determine $δ_{r,q}$ for all $q$ large enough, namely $q \geq \binom{r}{2}$. For example, we show that for $r \geq 4$, $δ_{r,q} = \frac{r}{r+1}$ for $\binom{r}{2} \leq q \leq \binom{r+1}{2}$ and $δ_{r,q} = \frac{r-1}{r}$ for $q \geq \binom{r+1}{2}+2$. Thus, $δ_{r,q}$ has a phase transition at $q = \binom{r+1}{2}$, where it drops from $\frac{r}{r+1}$ and then stabilizes at the existence threshold $\frac{r-1}{r}$. We also show that $δ_{r,q} \leq \frac{r}{r+1}$ for all $r,q$, supplementing and giving a new proof for the result of Balogh, Csaba, Pluhár and Treglown.
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Defect and transference versions of the Alon-Frankl-Lovasz theorem
Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac{n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon--Frankl--Lovász theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erdős matching conjecture.
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Is it easy to regularize a hypergraph with easy links?
A partition of a (hyper)graph is $\varepsilon$-homogenous if the edge densities between almost all clusters are either at most $\varepsilon$ or at least $1-\varepsilon$. Suppose a $3$-graph has the property that the link of every vertex has an $\varepsilon$-homogenous partition of size $\text{poly}(1/\varepsilon)$. Does this guarantee that the $3$-graph also has a small homogenous partition? Terry and Wolf proved that such a $3$-graph has an $\varepsilon$-homogenous partition of size given by a wowzer-type function. Terry recently improved this to a double exponential bound, and conjectured that this bound is tight. Our first result in this paper disproves this conjecture by giving an improved (single) exponential bound, which is best possible. We further obtain an analogous result for $k$-graphs of all uniformities $k \geq 3$. The above problem is part of a much broader programme which seeks to understand the conditions under which a (hyper)graph has small $\varepsilon$-regular partitions. While this problem is fairly well understood for graphs, the situation is (as always) much more involved already for $3$-graphs. For example, it is natural to ask if one can strengthen our first result by only requiring each link to have $\varepsilon$-regular partitions of size $\text{poly}(1/\varepsilon)$. Our second result shows that surprisingly the answer is `no', namely, a $3$-graph might only have regular partitions of tower-type size, even though the link of every vertex has an $\varepsilon$-regular partition of polynomial size.
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Subgraph discrepancies in the complete graph
Given a 2-edge-coloring $f : E(K_n) \rightarrow \{\pm 1\}$, the discrepancy of a subgraph $F \subseteq K_n$ is defined as $\left| \sum_{e \in E(F)} f(e) \right|$. Erdős, Füredi, Loebl and Sós showed that if $F$ is an $n$-vertex tree with maximum degree at most $(1-\varepsilon)n$, then every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $Ω(\varepsilon)n$. We extend this result by showing that the same conclusion holds for every $n$-vertex graph with maximum degree at most $(1-\varepsilon)n$ and no isolated vertices. We also show that for every $d$-regular $n$-vertex graph $F$ with $d \leq (1-\varepsilon)n$, every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $Ω(\sqrt{\varepsilon d}) \cdot n$. The dependence on $d$ and $n$ is best possible. Finally, we consider specific graphs $F$, namely $K_r$-factors and 2-factors. For each such graph $F$, we determine the optimal constant $λ$ such that every 2-coloring of $K_n$ has a copy of $F$ with discrepancy at least $(λ+ o(1))n$.
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Set mappings for general graphs
The study of extremal problems for set mappings has a long history. It was introduced in 1958 by Erdős and Hajnal, who considered the case of cliques in graphs and hypergraphs. Recently, Caro, Patkós, Tuza and Vizer revisited this subject, and initiated the systematic study of set mapping problems for general graphs. In this paper, we prove the following result, which answers one of their questions. Let $G$ be a graph with $m$ edges and no isolated vertices and let $f : E(K_N) \rightarrow E(K_N)$ such that $f(e)$ is disjoint from $e$ for all $e \in E(K_N)$. Then for some absolute constant $C$, as long as $N \geq C m$, there is a copy $G^*$ of $G$ in $K_N$ such that $f(e)$ is disjoint from $V(G^*)$ for all $e \in E(G^*)$. The bound $N = O(m)$ is tight for cliques and is tight up to a logarithmic factor for all $G$.
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Disperse Hypergraphs
For $\ell \geq 3$, an $\ell$-uniform hypergraph is disperse if the number of edges induced by any set of $\ell+1$ vertices is 0, 1, $\ell$ or $\ell+1$. We show that every disperse $\ell$-uniform hypergraph on $n$ vertices contains a clique or independent set of size $n^{Ω_{\ell}(1)}$, answering a question of the first author and Tomon. To this end, we prove several structural properties of disperse hypergraphs.
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The spanning tree spectrum: improved bounds and simple proofs
The number of spanning trees of a graph $G$, denoted $τ(G)$, is a well studied graph parameter with numerous connections to other areas of mathematics. In a recent remarkable paper, answering a question of Sedláček from 1969, Chan, Kontorovich and Pak showed that $τ(G)$ takes at least $1.1103^n$ different values across simple (and planar) $n$-vertex graphs $G$, for large enough $n$. We give a very short, purely combinatorial proof that at least $1.55^n$ values are attained. We also prove that exponential growth can be achieved with regular graphs, determining the growth rate in another problem first raised by Sedláček in the late 1960's. We further show that the following modular dual version of the result holds. For any integer $N$ and any $u < N$ there exists a planar graph on $O(\log N)$ vertices whose number of spanning trees is $u$ modulo $N$.
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Polynomial Property Testing
Property testers are fast, randomized "election polling"-type algorithms that determine if an input (e.g., graph or hypergraph) has a certain property or is $\varepsilon$-far from the property. In the dense graph model of property testing, it is known that many properties can be tested with query complexity that depends only on the error parameter $\varepsilon$ (and not on the size of the input), but the current bounds on the query complexity grow extremely quickly as a function of $1/\varepsilon$. Which properties can be tested efficiently, i.e., with $\mathrm{poly}(1/\varepsilon)$ queries? This survey presents the state of knowledge on this general question, as well as some key open problems.
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Regularity for hypergraphs with bounded VC$_2$ dimension
While Szemerédi's graph regularity lemma is an indispensable tool for studying extremal problems in graph theory, using it comes with a hefty price, since a worst-case graph may only have regular partitions of tower-type size. It is thus sensible to ask if there is some natural restriction which forces graphs to have much smaller regular partitions. A celebrated result of this type, due to Alon-Fischer-Newman and Lovász-Szegedy, states that for graphs of bounded VC dimension, one can reduce the tower-type bounds to polynomial. The graph regularity lemma has been extended to the setting of $k$-graphs by Gowers, Nagle-Rödl-Schacht-Skokan, and Tao. Unfortunately, these lemmas come with even larger Ackermann-type bounds. Chernikov-Starchenko and Fox-Pach-Suk considered a strong notion of $k$-graph VC dimension and proved that $k$-graphs of bounded VC dimension have regular partitions of polynomial size. Shelah introduced a weaker and combinatorially natural notion of dimension, called VC$_2$ dimension, which has since been extensively studied. In particular, Chernikov, Towsner, Terry, and Wolf asked if one can improve the worst case bounds for 3-graph regularity when the 3-graph has bounded VC$_2$ dimension. Our main result in this paper answers this question positively in the following strong sense: in the setting of bounded VC$_2$ dimension, one can reduce the bounds for 3-graph regularity by one level in Ackermann hierarchy. Furthermore, our new bound is best possible. Our proof has two key steps. We first introduce a new method for designing regularity lemmas for graphs of bounded VC dimension, based on the cylinder regularity lemma. We then prove a hypergraph version of the cylinder regularity lemma, which allows us to extend this method to hypergraphs. We also highlight a few other applications of this cylinder regularity lemma, which we expect to find many other uses.
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Steiner triple systems with high discrepancy
In this paper, we initiate the study of discrepancy questions for combinatorial designs. Specifically, we show that, for every fixed $r\ge 3$ and $n\equiv 1,3 \pmod{6}$, any $r$-colouring of the triples on $[n]$ admits a Steiner triple system of order $n$ with discrepancy $Ω(n^2)$. This is not true for $r=2$, but we are able to asymptotically characterise all $2$-colourings which do not contain a Steiner triple system with high discrepancy. The key step in our proofs is a characterization of 3-uniform hypergraphs avoiding a certain natural type of induced subgraphs, contributing to the structural theory of hypergraphs.
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Polynomial removal lemma for ordered matchings
We prove that for every ordered matching $H$ on $t$ vertices, if an ordered $n$-vertex graph $G$ is $\varepsilon$-far from being $H$-free, then $G$ contains $\text{poly}(\varepsilon) n^t$ copies of $H$. This proves a special case of a conjecture of Tomon and the first author. We also generalize this statement to uniform hypergraphs.
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Hypergraph removal with polynomial bounds
Given a fixed $k$-uniform hypergraph $F$, the $F$-removal lemma states that every hypergraph with few copies of $F$ can be made $F$-free by the removal of few edges. Unfortunately, for general $F$, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which $F$ one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when $F$ is $k$-partite. Alon proved that when $k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon's result can be extended to all $k>2$, namely, that the only $k$-graphs $F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when $F$ is $k$-partite. In this paper we prove this conjecture.
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