A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

Spiegel, Christoph

doi:10.37236/6730

Cited by 9 publications

(13 citation statements)

References 24 publications

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“…For readers familiar with the notion of partition and density regular (or invariant) matrices, note that they are trivially irredundant and positive. See [12] for an easy proof that they are also abundant. Finally, we introduce a parameter for abundant matrices originally due to Rödl and Ruciński [13].…”

Section: Generalized Van Der Waerden Gamesmentioning

confidence: 99%

Random Strategies are Nearly Optimal for Generalized van der Waerden Games

Kusch

Rué

Spiegel

et al. 2017

Electronic Notes in Discrete Mathematics

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In a (1 : q) Maker-Breaker game, one of the central questions is to find (or at least estimate) the maximal value of q that allows Maker to win the game. Based on the ideas of Bednarska and Luczak [1], who studied biased H-games, we prove general winning criteria for Maker and Breaker and a hypergraph generalization of their result. Furthermore, we study the biased version of a strong generalization of the van der Waerden games introduced by Beck [2] and apply our criteria to determine the threshold bias of these games up to constant factor. As in the result of [1], the random strategy for Maker is again the best known strategy.

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Section: Generalized Van Der Waerden Gamesmentioning

confidence: 99%

Random Strategies are Nearly Optimal for Generalized van der Waerden Games

Kusch

Rué

Spiegel

et al. 2017

Electronic Notes in Discrete Mathematics

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show abstract

“…Similarly, Schacht showed that for a given density regular matrix

A \in Z^{r \times m}

the threshold probability for the event that [ n ] p is ( δ , A )‐stable is

normalΘ false(n^{- 1 false/ m_{1} false(A false)} false)

. The third author as well as independently Hancock, Staden and Treglown extended this result to abundant matrices. Our result shows that the threshold bias of the Maker‐Breaker A ‐game lies around

normalΘ false(n^{1 false/ m_{1} false(A false)} false)

for the much broader class of positive and abundant matrices.…”

Section: Discussionmentioning

confidence: 84%

“…For readers familiar with the notion of partition and density regular (or invariant ) matrices in the homogeneous setting, note that they are trivially positive. See for an easy proof that they are also abundant..…”

Section: Introductionmentioning

confidence: 99%

On the optimality of the uniform random strategy

Kusch

Rué

Spiegel

et al. 2018

Random Struct Algorithms

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Biased Maker‐Breaker games, introduced by Chvátal and Erdős, are central to the field of positional games and have deep connections to the theory of random structures. The main questions are to determine the smallest bias needed by Breaker to ensure that Maker ends up with an independent set in a given hypergraph. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies certain “container‐type” regularity conditions. This will enable us to answer the main question for hypergraph generalizations of the H‐building games studied by Bednarska and Łuczak as well as a generalization of the van der Waerden games introduced by Beck. We find it remarkable that a purely game‐theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, while the analogous questions about sparse random discrete structures are usually quite challenging.

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“…Just before submitting the paper we were made aware of simultaneous and independent work of Spiegel [67]. In [67] the case r = 1 of Theorem 4.1 is proven. Spiegel also used the container method to give an alternative proof of Theorem 1.9.…”

Section: Introductionmentioning

confidence: 99%

Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers

Hancock¹,

Staden²,

Treglown³

2019

SIAM J. Discrete Math.

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Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij [3], and independently Saxton and Thomason [62] developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán properties) of (hyper)graphs and the integers. In particular:• We generalise the random Ramsey theorem of Rödl and Ruciński [54, 55, 56] by providing a resilience analogue. Our result unifies and generalises several fundamental results in the area including the random version of Turán's theorem due to Conlon and Gowers [14] and Schacht [64]. • The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter [40]. • All of the above results in fact hold for uniform hypergraphs.• For a (hyper)graph H, we determine, up to an error term in the exponent, the number of nvertex (hyper)graphs G that have the Ramsey property with respect to H (that is, whenever G is r-coloured, there is a monochromatic copy of H in G). • We strengthen the random Rado theorem of Friedgut, Rödl and Schacht [24] by proving a resilience version of the result. • For partition regular matrices A we determine, up to an error term in the exponent, the number of subsets of {1, . . . , n} for which there exists an r-colouring which contains no monochromatic solutions to Ax = 0. Along the way a number of open problems are posed. MSC2000: 5C30, 5C55, 5D10, 11B75.

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A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

Cited by 9 publications

References 24 publications

Random Strategies are Nearly Optimal for Generalized van der Waerden Games

Random Strategies are Nearly Optimal for Generalized van der Waerden Games

On the optimality of the uniform random strategy

Independent Sets in Hypergraphs and Ramsey Properties of Graphs and the Integers

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