Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

Cohen, Gil

doi:10.1137/16m1096219

Cited by 11 publications

(8 citation statements)

References 20 publications

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“…On the other hand, Erdős [14] famously used the probabilistic method to prove that, for all n, there exists an n-vertex graph with no homogeneous subgraph on 2 log 2 n vertices. Despite significant effort (see for example [5,9,8,23,28]), there are no known nonprobabilistic constructions of graphs whose largest homogeneous subgraphs are of a comparable size. Say an n-vertex graph is C-Ramsey if it has no homogeneous subgraph of size C log 2 n. It is widely believed that for any fixed constant C all C-Ramsey graphs must in some sense resemble random graphs, and this belief has been supported by a number of theorems showing that certain "richness" properties characteristic of random graphs hold for all C-Ramsey graphs.…”

Section: Ramsey Graphsmentioning

confidence: 99%

An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

Kwan,

Sauermann

2019

Preprint

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Consider a quadratic polynomial f (ξ1, . . . , ξn) of independent Bernoulli random variables. What can be said about the concentration of f on any single value? This generalises the classical Littlewood-Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of f can be as large as about 1/√ n, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features f must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if f has point probabilities much larger than 1/n then it must be close to a quadratic form with low rank. We also give an application to Ramsey graphs, asymptotically answering a question of Kwan, Sudakov and Tran.

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Section: Ramsey Graphsmentioning

confidence: 99%

An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

Kwan,

Sauermann

2019

Preprint

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“…Until very recently, the best explicit two-source extractor due to Bourgain [Bou05] required min-entropy at least k ≥ cn for some constant c = 0.49... < 0.5; the best explicit two-source disperser due to [BRSW12] required min-entropy at least exp(poly(log log n)). Chattopadhyay and Zuckerman broke the barrier for two-source extractors and gave an explicit construction for min-entropies at least C(log n) 74 ; independently, Cohen [Coh15] gave an explicit two-source disperser for min-entropy log C n for some (unspecified) constant C. We show the following: Theorem 1.4. For a sufficiently big constant C, there exists an explicit (n, k) two-source extractor D : {0, 1} n × {0, 1} n → {0, 1} with constant-error for k ≥ C log 10 n. In particular, we get an explicit (n, k) two-source disperser for k ≥ C log 10 n.…”

Section: Two-source Extractorsmentioning

confidence: 85%

Explicit Resilient Functions Matching Ajtai-Linial

Meka¹

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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A Boolean function on n variables is q-resilient if for any subset of at most q variables, the function is very likely to be determined by a uniformly random assignment to the remaining n − q variables; in other words, no coalition of at most q variables has significant influence on the function. Resilient functions have been extensively studied with a variety of applications in cryptography, distributed computing, and pseudorandomness. The best known resilient function on n variables due to Ajtai and Linial [AL93] has the property that only sets of size Ω(n/(log 2 n)) can have influence bounded away from zero. However, the construction of Ajtai and Linial is by the probabilistic method and does not give an efficiently computable function.We construct an explicit monotone depth three almostbalanced Boolean function on n bits that is Ω(n/(log 2 n))-resilient matching the bounds of Ajtai and Linial. The best previous explicit constructions of Meka [Mek09] (which only gives a logarithmic depth function), and Chattopadhyay and Zuckerman [CZ15] were only (n 1−β )-resilient for any constant 0 < β < 1. Our construction and analysis are motivated by (and simplifies parts of) the recent breakthrough of [CZ15] giving explicit two-sources extractors for polylogarithmic min-entropy; a key ingredient in their result was the construction of explicit constant-depth resilient functions.An important ingredient in our construction is a new randomness-optimal oblivious sampler that preserves moment generating functions of sums of variables and could be useful elsewhere.

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“…Such graphs are strongly-algebraic of complexity (1, (p − 1)/2), and it is widely believed that Paley-graphs have only polylogarithmic sized cliques or independent sets. For the best known explicit constructions of Ramsey graphs, see the recent works of Chattopadhyay, Zuckerman [7] and Cohen [11].…”

Section: Ramsey Properties Of Algebraic Graphsmentioning

confidence: 99%

Ramsey properties of algebraic graphs and hypergraphs

Sudakov¹,

Tomon²

2021

Preprint

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One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on N vertices. By the celebrated result of Erdős from 1947, the random graph on N vertices with edge probability 1/2, contains no clique or independent set larger than 2 log 2 N , with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools.Say that an r-uniform hypergraph H is algebraic of complexity (n, d, m) if the vertices of H are elements of F n for some field F, and there exist m polynomials f 1 , . . . , f m : (F n ) r → F of degree at most d such that the edges of H are determined by the zero-patterns of f 1 , . . . , f m . The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity (n, d, m) has good Ramsey properties, then at least one of the parameters n, d, m must be large.In 2001, Rónyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if G is an algebraic graph of complexity (n, d, m) on N vertices, then either G or its complement contains a complete balanced bipartite graph of size Ω n,d,m (N 1/(n+1) ). We extend this result by showing that such G contains either a clique or an independent set of size N Ω(1/ndm) and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for r-uniform algebraic hypergraphs that are defined by a single polynomial, that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.

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Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

Cited by 11 publications

References 20 publications

An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

Explicit Resilient Functions Matching Ajtai-Linial

Ramsey properties of algebraic graphs and hypergraphs

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