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14
Extremal results in sparse pseudorandom graphs
 ADV. MATH. 256 (2014), 206–290
, 2014
"... Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extendin ..."
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Cited by 13 (8 self)
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Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi’s regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a wellknown open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several wellknown combinatorial theorems, including the removal lemmas for graphs and groups, the ErdősStoneSimonovits theorem and Ramsey’s
AN Lp THEORY OF SPARSE GRAPH CONVERGENCE I: LIMITS, SPARSE RANDOM GRAPH MODELS, AND POWER LAW
"... Abstract. We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L ∞ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots ..."
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Cited by 10 (2 self)
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Abstract. We introduce and develop a theory of limits for sequences of sparse graphs based on Lp graphons, which generalizes both the existing L ∞ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the Lp theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper.
Graph removal lemmas
 SURVEYS IN COMBINATORICS
, 2013
"... The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and com ..."
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Cited by 9 (3 self)
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The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.
Testing low complexity affineinvariant properties
, 2013
"... Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a ..."
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Cited by 8 (3 self)
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Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the ReedMuller code over F p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that low degree is preserved by composition with affine maps. The complexity of an affineinvariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affineinvariant property P of functions f : F n p → [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higherorder Fourier analysis.
Graphs with few 3cliques and 3anticliques are 3universal
 J. GRAPH THEORY
, 2014
"... For given integers k, l we ask whether every large graph with a sufficiently small number of kcliques and kanticliques must contain an induced copy of every lvertex graph. Here we prove this claim for k = l = 3 with a sharp bound. A similar phenomenon is established as well ..."
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Cited by 4 (2 self)
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For given integers k, l we ask whether every large graph with a sufficiently small number of kcliques and kanticliques must contain an induced copy of every lvertex graph. Here we prove this claim for k = l = 3 with a sharp bound. A similar phenomenon is established as well
A Short Proof of Gowers ’ Lower Bound for the Regularity Lemma
"... A celebrated result of Gowers states that for every ɛ> 0 there is a graph G such that every ɛregular partition of G (in the sense of Szemerédi’s regularity lemma) has order given by a tower of exponents of height polynomial in 1/ɛ. In this note we give a new proof of this result that uses a cons ..."
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Cited by 3 (1 self)
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A celebrated result of Gowers states that for every ɛ> 0 there is a graph G such that every ɛregular partition of G (in the sense of Szemerédi’s regularity lemma) has order given by a tower of exponents of height polynomial in 1/ɛ. 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. 1
A characterization of locally testable affineinvariant properties via decomposition theorems
 In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC
, 2014
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Density and regularity theorems for semialgebraic hypergraphs
, 2014
"... A kuniform semialgebraic hypergraph H is a pair (P,E), where P is a subset of Rd and E is a collection of ktuples {p1,..., pk} ⊂ P such that (p1,..., pk) ∈ E if and only if the kd coordinates of the pis satisfy a boolean combination of a finite number of polynomial inequalities. The complexity ..."
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A kuniform semialgebraic hypergraph H is a pair (P,E), where P is a subset of Rd and E is a collection of ktuples {p1,..., pk} ⊂ P such that (p1,..., pk) ∈ E if and only if the kd coordinates of the pis satisfy a boolean combination of a finite number of polynomial inequalities. The complexity of H can be measured by the number and the degrees of these inequalities and the number of variables (coordinates) kd. Several classical results in extremal hypergraph theory can be substantially improved when restricted to semialgebraic hypergraphs. Substantially improving a theorem of Fox, Gromov, Lafforgue, Naor, and Pach, we establish the following “polynomial regularity lemma”: For any 0 < ε < 1/2, the vertex set of every kuniform semialgebraic hypergraph H = (P,E) can be partitioned into at most (1/ε)c parts P1, P2,..., as equal as possible, such that all but an at most εfraction of the ktuples of parts (Pi1,..., Pik) are homogeneous in the sense that either every ktuple (pi1,..., pik) ∈ Pi1×...×Pik belongs to E or none of them do. Here c> 0 is a constant that depends on the complexity of H. We also establish an improved lower bound, single exponentially decreasing in k, on the best constant δ> 0 such that the vertex classes P1,..., Pk of every kpartite kuniform semialgebraic