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A linear equation $E$ is said to be sparse if there is $c>0$ so that every subset of $[n]$ of size $n^{1-c}$ contains a solution of $E$ in distinct integers. The problem of characterizing the sparse equations, first raised by Ruzsa in the 90's, is one of the most important open problems in additive combinatorics. We say that $E$ in $k$ variables is abundant if every subset of $[n]$ of size $\varepsilon n$ contains at least poly$(\varepsilon)\cdot n^{k-1}$ solutions of $E$. It is clear that every abundant $E$ is sparse, and Gir\~{a}o, Hurley, Illingworth and Michel asked if the converse implication also holds. In this note we show that this is the case for every $E$ in $4$ variables. We further discuss a generalization of this problem which applies to all linear equations.

Graph modification problems ask for the minimal number of vertex/edge additions/deletions needed to make a graph satisfy some predetermined property. A (meta) problem of this type, which was raised by Yannakakis in 1981, asks to determine for which properties ${\mathcal P}$, it is NP-hard to compute the smallest number of edge deletions needed to make a graph satisfy ${\mathcal P}$. Despite being extensively studied in the past 40 years, this problem is still wide open. In fact, it is open even when ${\mathcal P}$ is the property of being $H$-free, for some fixed graph $H$. In this case we use $\text{rem}_{H}(G)$ to denote the smallest number of edge deletions needed to turn $G$ into an $H$-free graph. Alon, Sudakov and Shapira [Annals of Math. 2009] proved that if $H$ is not bipartite, then computing $\text{rem}_{H}(G)$ is NP-hard. They left open the problem of classifying the bipartite graphs $H$ for which computing $\text{rem}_{H}(G)$ is NP-hard. In this paper we resolve this problem when $H$ is a forest, showing that computing $\text{rem}_{H}(G)$ is polynomial-time solvable if $H$ is a star forest and NP-hard otherwise. Our main innovation in this work lies in introducing a new graph theoretic approach for Yannakakis's problem, which differs significantly from all prior works on this subject. In particular, we prove new results concerning an old and famous conjecture of Erd\H{o}s and S\'os, which are of independent interest.

A theorem of R\"odl states that for every fixed $F$ and $\varepsilon>0$ there is $\delta=\delta_F(\varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $\delta n$ whose edge density is either at most $\varepsilon$ or at least $1-\varepsilon$. R\"odl's proof relied on the regularity lemma, hence it supplied only a tower-type bound for $\delta$. Fox and Sudakov conjectured that $\delta$ can be made polynomial in $\varepsilon$, and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when $F=P_4$. In fact, they show that the same conclusion holds even if $G$ contains few copies of $P_4$. In this note we give a short proof of a more general statement.

A distance estimator for a graph property $\mathcal{P}$ is an algorithm that given $G$ and $\alpha, \varepsilon >0$ distinguishes between the case that $G$ is $(\alpha-\varepsilon)$-close to $\mathcal{P}$ and the case that $G$ is $\alpha$-far from $\mathcal{P}$ (in edit distance). We say that $\mathcal{P}$ is estimable if it has a distance estimator whose query complexity depends only on $\varepsilon$. Every estimable property is also testable, since testing corresponds to estimating with $\alpha=\varepsilon$. A central result in the area of property testing, the Fischer--Newman theorem, gives an inverse statement: every testable property is in fact estimable. The proof of Fischer and Newman was highly ineffective, since it incurred a tower-type loss when transforming a testing algorithm for $\mathcal{P}$ into a distance estimator. This raised the natural problem, studied recently by Fiat--Ron and by Hoppen--Kohayakawa--Lang--Lefmann--Stagni, whether one can find a transformation with a polynomial loss. We obtain the following results. 1. If $\mathcal{P}$ is hereditary, then one can turn a tester for $\mathcal{P}$ into a distance estimator with an exponential loss. This is an exponential improvement over the result of Hoppen et. al., who obtained a transformation with a double exponential loss. 2. For every $\mathcal{P}$, one can turn a testing algorithm for $\mathcal{P}$ into a distance estimator with a double exponential loss. This improves over the transformation of Fischer--Newman that incurred a tower-type loss. Our main conceptual contribution in this work is that we manage to turn the approach of Fischer--Newman, which was inherently ineffective, into an efficient one. On the technical level, our main contribution is in establishing certain properties of Frieze--Kannan Weak Regular partitions that are of independent interest.

The celebrated Brown-Erd\H{o}s-S\'os conjecture states that for every fixed $e$, every $3$-uniform hypergraph with $\Omega(n^2)$ edges contains $e$ edges spanned by $e+3$ vertices. Up to this date all the approaches towards resolving this problem relied on highly involved applications of the hypergraph regularity method, and yet they supplied only approximate versions of the conjecture, producing $e$ edges spanned by $e+O(\log e/\log \log e)$ vertices. In this short paper we describe a completely different approach, which reduces the problem to a variant of another well-known conjecture in extremal graph theory. A resolution of the latter would resolve the Brown-Erd\H{o}s-S\'os conjecture up to an absolute additive constant.

The triangle removal states that if $G$ contains $\varepsilon n^2$ edge-disjoint triangles, then $G$ contains $\delta(\varepsilon)n^3$ triangles. Unfortunately, there are no sensible bounds on the order of growth of $\delta(\varepsilon)$, and at any rate, it is known that $\delta(\varepsilon)$ is not polynomial in $\varepsilon$. Csaba recently obtained an asymmetric variant of the triangle removal, stating that if $G$ contains $\varepsilon n^2$ edge-disjoint triangles, then $G$ contains $2^{-\mathrm{poly}(1/\varepsilon)}\cdot n^5$ copies of $C_5$. To this end, he devised a new variant of Szemer\'edi's regularity lemma. We obtain the following results: - We first give a regularity-free proof of Csaba's theorem, which improves the number of copies of $C_5$ to the optimal number $\mathrm{poly}(\varepsilon)\cdot n^5$. - We say that $H$ is $K_3$-abundant if every graph containing $\varepsilon n^2$ edge-disjoint triangles has $\mathrm{poly}(\varepsilon)\cdot n^{|V(H)|}$ copies of $H$. It is easy to see that a $K_3$-abundant graph must be triangle-free and tripartite. Given our first result, it is natural to ask if all triangle-free tripartite graphs are $K_3$-abundant. Our second result is that assuming a well-known conjecture of Ruzsa in additive number theory, the answer to this question is negative. Our proofs use a mix of combinatorial, number-theoretic, probabilistic, and Ramsey-type arguments., Comment: 20 pages

Let $N_{\mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$ vertex planar graph. The problem of bounding this function for various graphs $H$ has been extensively studied since the 70's. A special case that received a lot of attention recently is when $H$ is the path on $2m+1$ vertices, denoted $P_{2m+1}$. Our main result in this paper is that $$ N_{\mathcal{P}}(n,P_{2m+1})=O(m^{-m}n^{m+1})\;.$$ This improves upon the previously best known bound by a factor $e^{m}$, which is best possible up to the hidden constant, and makes a significant step towards resolving conjectures of Gosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.

Let $Q(n)$ denote the number of integers $1 \leq q \leq n$ whose prime factorization $q= \prod^{t}_{i=1}p^{a_i}_i$ satisfies $a_1\geq a_2\geq \ldots \geq a_t$. Hardy and Ramanujan proved that $$ \log Q(n) \sim \frac{2\pi}{\sqrt{3}} \sqrt{\frac{\log(n)}{\log\log(n)}}\;. $$ Before proving the above precise asymptotic formula, they studied in great detail what can be obtained concerning $Q(n)$ using purely elementary methods, and were only able to obtain much cruder lower and upper bounds using such methods. In this paper we show that it is in fact possible to obtain a purely elementary (and much shorter) proof of the Hardy--Ramanujan Theorem. Towards this goal, we first give a simple combinatorial argument, showing that $Q(n)$ satisfies a (pseudo) recurrence relation. This enables us to replace almost all the hard analytic part of the original proof with a short inductive argument.

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 R\"odl 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.

Given a fixed hypergraph $H$, let $\mbox{wsat}(n,H)$ denote the smallest number of edges in an $n$-vertex hypergraph $G$, with the property that one can sequentially add the edges missing from $G$, so that whenever an edge is added, a new copy of $H$ is created. The study of $\mbox{wsat}(n,H)$ was introduced by Bollob\'as in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most $H$ very little is known regarding $\mbox{wsat}(n,H)$, Alon proved in 1985 that for every graph $H$ there is a limiting constant $C_H$ so that $\mbox{wsat}(n,H)=(C_H+o(1))n$. Tuza conjectured in 1992 that Alon's theorem can be (appropriately) extended to arbitrary $r$-uniform hypergraphs. In this paper we prove this conjecture.

For a set of positive integers $A$, let $p_A(n)$ denote the number of ways to write $n$ as a sum of integers from $A$, and let $p(n)$ denote the usual partition function. In the early 40s, Erd\H{o}s extended the classical Hardy--Ramanujan formula for $p(n)$ by showing that $A$ has density $\alpha$ if and only if $\log p_A(n) \sim \log p(\alpha n)$. Nathanson asked if Erd\H{o}s's theorem holds also with respect to $A$'s lower density, namely, whether $A$ has lower-density $\alpha$ if and only if $\log p_A(n) / \log p(\alpha n)$ has lower limit $1$. We answer this question negatively by constructing, for every $\alpha > 0$, a set of integers $A$ of lower density $\alpha$, satisfying $$ \liminf_{n \rightarrow \infty} \frac{\log p_A(n)}{\log p(\alpha n)} \geq \left(\frac{\sqrt{6}}{\pi}-o_{\alpha}(1)\right)\log(1/\alpha)\;. $$ We further show that the above bound is best possible (up to the $o_\alpha(1)$ term), thus determining the exact extremal relation between the lower density of a set of integers and the lower limit of its partition function. We also prove an analogous theorem with respect to the upper density of a set of integers, answering another question of Nathanson.

For fixed $m$ and $R\subseteq \{0,1,\ldots,m-1\}$, take $A$ to be the set of positive integers congruent modulo $m$ to one of the elements of $R$, and let $p_A(n)$ be the number of ways to write $n$ as a sum of elements of $A$. Nathanson proved that $\log p_A(n) \leq (1+o(1)) \pi \sqrt{2n|R|/3m}$ using a variant of a remarkably simple method devised by Erd\H{o}s in order to bound the partition function. In this short note we describe a simpler and shorter proof of Nathanson's bound., Comment: To appear in Amer. Math. Monthly

The pantograph differential equation and its solution, the deformed exponential function, are remarkable objects that appear in areas as diverse as combinatorics, number theory, statistical mechanics, and electrical engineering. In this article we describe a new surprising application of these objects in graph theory, by showing that the set of all cliques is not forcing for quasirandomness. This provides a natural example of an infinite family of graphs, which is not forcing, and answers a natural question posed by P. Horn., Comment: To appear in Amer. Math. Monthly

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80's, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: 1. One can compute the number of homomorphic copies of $C_{2k}$ and $C_{2k+1}$ in $n$-vertex graphs of bounded degeneracy in time $\tilde{O}(n^{d_{k}})$, where the fastest known algorithm for detecting directed copies of $C_k$ in general $m$-edge digraphs runs in time $\tilde{O}(m^{d_{k}})$. 2. Conversely, one can transform any $O(n^{b_{k}})$ algorithm for computing the number of homomorphic copies of $C_{2k}$ or of $C_{2k+1}$ in $n$-vertex graphs of bounded degeneracy, into an $\tilde{O}(m^{b_{k}})$ time algorithm for detecting directed copies of $C_k$ in general $m$-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of $C_k$-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams.

We show that there exist linear $3$-uniform hypergraphs with $n$ vertices and $\Omega(n^2)$ edges which contain no copy of the $3 \times 3$ grid. This makes significant progress on a conjecture of F\"{u}redi and Ruszink\'{o}. We also discuss connections to proving lower bounds for the $(9,6)$ Brown-Erd\H{o}s-S\'{o}s problem and to a problem of Solymosi and Solymosi.

We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that $H$ is easy if there is a linear-time algorithm for counting the number of copies of $H$ in an input $G$ of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every $H$ on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all $H$ on 5 vertices, and further proved that for every $k > 5$ there is a $k$-vertex $H$ which is not easy. They left open the natural problem of characterizing all easy graphs $H$. Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph $H$ to be easy. Here we show that this sufficient condition is also necessary, thus fully answering the Bera--Pashanasangi--Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.

Property testers are fast randomized algorithms whose task is to distinguish between inputs satisfying some predetermined property ${\cal P}$ and those that are far from satisfying it. Since these algorithms operate by inspecting a small randomly selected portion of the input, the most natural property one would like to be able to test is whether the input does not contain certain forbidden small substructures. In the setting of graphs, such a result was obtained by Alon et al., who proved that for any finite family of graphs ${\cal F}$, the property of being induced ${\cal F}$-free (i.e. not containing an induced copy of any $F \in {\cal F}$) is testable. It is natural to ask if one can go one step further and prove that more elaborate properties involving induced subgraphs are also testable. One such generalization of the result of Alon et al. was formulated by Goldreich and Shinkar who conjectured that for any finite family of graphs ${\cal F}$, and any linear inequality involving the densities of the graphs $F \in {\cal F}$ in the input graph, the property of satisfying this inequality can be tested in a certain restricted model of graph property testing. Our main result in this paper disproves this conjecture in the following strong form: some properties of this type are not testable even in the classical (i.e. unrestricted) model of graph property testing. The proof deviates significantly from prior non-testability results in this area. The main idea is to use a linear inequality relating induced subgraph densities in order to encode the property of being a quasirandom graph.

Let $f(n,v,e)$ denote the maximum number of edges in a $3$-uniform hypergraph not containing $e$ edges spanned by at most $v$ vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges $e \geq 3$, what is the smallest integer $d=d(e)$ so that $f(n,e+d,e) = o(n^2)$? This question has its origins in work of Brown, Erd\H{o}s and S\'os from the early 70's and the standard conjecture is that $d(e)=3$ for every $e \geq 3$. The state of the art result regarding this problem was obtained in 2004 by S\'{a}rk\"{o}zy and Selkow, who showed that $f(n,e + 2 + \lfloor \log_2 e \rfloor,e) = o(n^2)$. The only improvement over this result was a recent breakthrough of Solymosi and Solymosi, who improved the bound for $d(10)$ from 5 to 4. We obtain the first asymptotic improvement over the S\'{a}rk\"{o}zy--Selkow bound, showing that $$ f(n, e + O(\log e/ \log\log e), e) = o(n^2). $$

An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-Erd\H{o}s-S\'os conjecture states that for every fixed $k$ and $r$, every linear $r$-graph with $\Omega(n^2)$ edges contains $k$ edges spanned by at most $(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely, that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete linear $r$-graph, one can find $k$ monochromatic edges spanned by at most $(r-2)k+3$ vertices. We prove that this Ramsey version of the conjecture holds under the additional assumption that $r \geq r_0(c)$, and we show that for $c=2$ it holds for all $r\geq 4$.

The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-R\"odl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers' famous lower bound for graph regularity., Comment: To appear in GAFA. This is a merged version of arXiv:1804.05511 and arXiv:1804.05513. See arXiv:1804.05511 for a self contained proof of the theorem for the special case of 3-uniform hypergraphs

The area of graph property testing seeks to understand the relation between the global properties of a graph and its local statistics. In the classical model, the local statistics of a graph is defined relative to a uniform distribution over the graph's vertex set. A graph property $\mathcal{P}$ is said to be testable if the local statistics of a graph can allow one to distinguish between graphs satisfying $\mathcal{P}$ and those that are far from satisfying it. Goldreich recently introduced a generalization of this model in which one endows the vertex set of the input graph with an arbitrary and unknown distribution, and asked which of the properties that can be tested in the classical model can also be tested in this more general setting. We completely resolve this problem by giving a (surprisingly "clean") characterization of these properties. To this end, we prove a removal lemma for vertex weighted graphs which is of independent interest., Comment: A short version of this paper appeared in the Proceedings of STOC '19

A well known observation of Lov\'asz is that if a hypergraph is not $2$-colorable, then at least one pair of its edges intersect at a single vertex. %This very simple criterion turned out to be extremly useful . In this short paper we consider the quantitative version of Lov\'asz's criterion. That is, we ask how many pairs of edges intersecting at a single vertex, should belong to a non $2$-colorable $n$-uniform hypergraph? Our main result is an {\em exact} answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollob\'as's two families theorem with Pluhar's randomized coloring algorithm., Comment: A note on Property B

The Erd\H{o}s--Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph $H$, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of $r$-uniform hypergraphs. A theorem of R\"odl asserts that if an $n$-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to $0$ or $1$) of size $\Omega(n)$. We prove that if a $3$-uniform hypergraph is non-universal then it contains an almost homogeneous set of size $\Omega(\log n)$. An example of R\"odl from 1986 shows that this bound is tight. Let $R_r(t)$ denote the size of the largest non-universal $r$-graph $G$ so that neither $G$ nor its complement contain a complete $r$-partite subgraph with parts of size $t$. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma, showing how to transform a lower bound for $R_{r}(t)$ into a lower bound for $R_{r+1}(t)$. As an application of this lemma, we improve a bound of Conlon--Fox--Sudakov by showing that $R_3(t) \geq t^{\Omega(t)}$., Comment: 19 pages

The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-R\"odl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming a prediction of Tao. Prior to our work, the only result of the above type was Gowers' famous lower bound for graph regularity. In this paper we describe the key new ideas which enable us to overcome several barriers which stood in the way of establishing such bounds for hypergraphs of higher uniformity. One of them is a tight bound for a new (very weak) version of the {\em graph} regularity lemma. Using this bound, we prove a lower bound for any regularity lemma of $3$-uniform hypergraphs that satisfies certain mild conditions. We then show how to use this result in order to prove a tight bound for the hypergraph regularity lemmas of Gowers and Frankl--R\"odl. We will obtain similar results for hypergraphs of arbitrary uniformity $k\geq 2$ in a subsequent paper., Comment: This manuscript contains the proof of the main result of arXiv:1907.07639 when specialized to 3-uniform hypergraphs

The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ${\ell}$ and $k$, how many copies of the $k$-cycle guarantee the appearance of an $\ell$-cycle? Extending previous results of Bollob\'as--Gy\H{o}ri--Li and Alon--Shikhelman, we fully resolve this problem by giving tight (or nearly tight) bounds for all values of $\ell$ and $k$. We also present a somewhat surprising application of the above mentioned estimates to the study of the graph removal lemma. Prior to this work, all bounds for removal lemmas were either polynomial or there was a tower-type gap between the best known upper and lower bounds. We fill this gap by showing that for every super-polynomial function $f(\varepsilon)$, there is a family of graphs ${\cal F}$, such that the bounds for the ${\cal F}$ removal lemma are precisely given by $f(\varepsilon)$. We thus obtain the first examples of removal lemmas with tight super-polynomial bounds. A special case of this result resolves a problem of Alon and the second author, while another special case partially resolves a problem of Goldreich.

Suppose one needs to change the direction of at least $\epsilon n^2$ edges of an $n$-vertex tournament $T$, in order to make it $H$-free. A standard application of the regularity method shows that in this case $T$ contains at least $f^*_H(\epsilon)n^h$ copies of $H$, where $f^*_H$ is some tower-type function. It has long been observed that many graph/digraph problems become easier when assuming that the host graph is a tournament. It is thus natural to ask if the removal lemma becomes easier if we assume that the digraph $G$ is a tournament. Our main result here is a precise characterization of the tournaments $H$ for which $f^*_H(\epsilon)$ is polynomial in $\epsilon$, stating that such a bound is attainable if and only if $H$'s vertex set can be partitioned into two sets, each spanning an acyclic directed graph. The proof of this characterization relies, among other things, on a novel application of a regularity lemma for matrices due to Alon, Fischer and Newman, and on probabilistic variants of Ruzsa-Szemer\'edi graphs. We finally show that even when restricted to tournaments, deciding if $H$ satisfies the condition of our characterization is an NP-hard problem.

Obtaining an efficient bound for the triangle removal lemma is one of the most outstanding open problems of extremal combinatorics. Perhaps the main bottleneck for achieving this goal is that triangle-free graphs can be highly unstructured. For example, triangle-free graphs might have only regular partitions (in the sense of Szemer\'edi) of tower-type size. And indeed, essentially all the graph properties ${\cal P}$ for which removal lemmas with reasonable bounds were obtained, are such that every graph satisfying ${\cal P}$ has a small regular partition. So in some sense, a barrier for obtaining an efficient removal lemma for property ${\cal P}$ was having an efficient regularity lemma for graphs satisfying ${\cal P}$. In this paper we consider the property of being induced $C_4$-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for the induced $C_4$ removal lemma. We thus obtain the first efficient removal lemma that does not rely on an efficient version of the regularity lemma. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox.

A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. One of the most celebrated results of this type is the Ruzsa-Szemer\'edi triangle removal lemma, which states that if a graph is $\varepsilon$-far from being triangle free, then most subsets of vertices of size $C(\varepsilon)$ are not triangle free. Unfortunately, the best known upper bound on $C(\varepsilon)$ is given by a tower-type function, and it is known that $C(\varepsilon)$ is not polynomial in $\varepsilon^{-1}$. The triangle removal lemma has been extended to many other graph properties, and for some of them the corresponding function $C(\varepsilon)$ is polynomial. This raised the natural question, posed by Goldreich in 2005 and more recently by Alon and Fox, of characterizing the properties for which one can prove removal lemmas with polynomial bounds. Our main results in this paper are new sufficient and necessary criteria for guaranteeing that a graph property admits a removal lemma with a polynomial bound. Although both are simple combinatorial criteria, they imply almost all prior positive and negative results of this type. Moreover, our new sufficient conditions allow us to obtain polynomially bounded removal lemmas for many properties for which the previously known bounds were of tower-type. In particular, we show that every {\em semi-algebraic} graph property admits a polynomially bounded removal lemma. This confirms a conjecture of Alon.

We introduce a new variant of Szemer\'edi's regularity lemma which we call the "sparse regular approximation lemma" (SRAL). The input to this lemma is a graph $G$ of edge density $p$ and parameters $\epsilon, \delta$, where we think of $\delta$ as a constant. The goal is to construct an $\epsilon$-regular partition of $G$ while having the freedom to add/remove up to $\delta |E(G)|$ edges. As we show here, this weaker variant of the regularity lemma already suffices for proving the graph removal lemma and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma? Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height $O(\log 1/p)$. This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height $\Omega(\log 1/p)$ is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.

We consider the following Tur\'an-type problem: given a fixed tournament $H$, what is the least integer $t=t(n,H)$ so that adding $t$ edges to any $n$-vertex tournament, results in a digraph containing a copy of $H$. Similarly, what is the least integer $t=t(T_n,H)$ so that adding $t$ edges to the $n$-vertex transitive tournament, results in a digraph containing a copy of $H$. Besides proving several results on these problems, our main contributions are the following: (1) Pach and Tardos conjectured that if $M$ is an acyclic $0/1$ matrix, then any $n \times n$ matrix with $n(\log n)^{O(1)}$ entries equal to $1$ contains the pattern $M$. We show that this conjecture is equivalent to the assertion that $t(T_n,H)=n(\log n)^{O(1)}$ if and only if $H$ belongs to a certain (natural) family of tournaments. (2) We propose an approach for determining if $t(n,H)=n(\log n)^{O(1)}$. This approach combines expansion in sparse graphs, together with certain structural characterizations of $H$-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach-Tardos conjecture.

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge $e$ in $G$ we would like to decide whether $e$ belongs to a connected subgraph $G'$ consisting of $(1+\epsilon)n$ edges (for a prespecified constant $\epsilon >0$), where the decision for different edges should be consistent with the same subgraph $G'$. Can this task be performed by inspecting only a {\em constant} number of edges in $G$? Our main results are: (1) We show that if every $t$-vertex subgraph of $G$ has expansion $1/(\log t)^{1+o(1)}$ then one can (deterministically) construct a sparse spanning subgraph $G'$ of $G$ using few inspections. To this end we analyze a "local" version of a famous minimum-weight spanning tree algorithm. (2) We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of $3$-regular graphs of high girth, in which every $t$-vertex subgraph has expansion $1/(\log t)^{1-o(1)}$., Comment: References fixed

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. These results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{\'e}di regularity lemma in graph theory. It shows that for any abelian group $G$ and any bounded function $f:G \to [0,1]$, there exists a subgroup $H \le G$ of bounded index such that, when restricted to most cosets of $H$, the function $f$ is pseudorandom in the sense that all its nontrivial Fourier coefficients are small. Quantitatively, if one wishes to obtain that for $1-\epsilon$ fraction of the cosets, the nontrivial Fourier coefficients are bounded by $\epsilon$, then Green shows that $|G/H|$ is bounded by a tower of twos of height $1/\epsilon^3$. He also gives an example showing that a tower of height $\Omega(\log 1/\epsilon)$ is necessary. Here, we give an improved example, showing that a tower of height $\Omega(1/\epsilon)$ is necessary.

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is ``close'' to a graph $G'$ with $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree. Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.

A celebrated result of Gowers states that for every \epsilon > 0 there is a graph G so that every \epsilon-regular partition of G (in the sense of Szemeredi's regularity lemma) has order given by a tower of exponents of height polynomial in 1/\epsilon. In this note we give a new proof of this result that uses a construction and proof of correctness that are significantly simpler and shorter.

A graph property P is said to be testable if one can check if a graph is close or far from satisfying P using few random local inspections. Property P is said to be non-deterministically testable if one can supply a "certificate" to the fact that a graph satisfies P so that once the certificate is given its correctness can be tested. The notion of non-deterministic testing of graph properties was recently introduced by Lovasz and Vesztergombi, who proved that (somewhat surprisingly) a graph property is testable if and only if it is non-deterministically testable. Their proof used graph limits, and so it did not supply any explicit bounds. They thus asked if one can obtain a proof of their result which will supply such bounds. We answer their question positively by proving their result using Szemeredi's regularity lemma. An interesting aspect of our proof is that it highlights the fact that the regularity lemma can be interpreted as saying that all graphs can be approximated by finitely many "template" graphs.