Counting configuration–free sets in groups

Rué, Juanjo; Serra, Oriol; Vena, Lluís

doi:10.1016/j.endm.2015.06.075

Cited by 3 publications

(4 citation statements)

References 35 publications

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“…Together with (8), we see that (10), (11) and ( 12) imply that only s-milky ways contribute meaningfully when s ≤ ⌊k/2⌋. Sadly, this bound is not quite strong enough when s > ⌊k/2⌋, since s-milky ways do not exist here.…”

Section: And We See By Induction Hypothesis Thatmentioning

confidence: 80%

“…These ideas, named hypergraph container method were then used by Spiegel [15] and independently by Hancock, Staden and Treglown [6] to extend [5,Theorem 2] to the broadest class of linear systems possible. Similar techniques were used by Rué, Serra and Vena [10] to study random sparse analogues of [5,Theorem 2] in finite fields and more general configurations than linear systems of equations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Normal limiting distributions for systems of linear equations in random sets

Rué,

Wötzel

2021

Preprint

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We consider the binomial random set model [n] p where each element in {1, . . . , n} is chosen independently with probability p := p(n). We show that for essentially all regimes of p and very general conditions for a matrix A and a column vector b, the count of specific integer solutions to the system of linear equations Ax = b with the entries of x in [n] p follows a (conveniently rescaled) normal limiting distribution. This applies among others to the number of solutions with every variable having a different value, as well as to a broader class of so-called non-trivial solutions in homogeneous strictly balanced systems. Our proof relies on the delicate linear algebraic study both of the subjacent matrices and the corresponding ranks of certain submatrices, together with the application of the method of moments in probability theory.

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Section: And We See By Induction Hypothesis Thatmentioning

confidence: 80%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Normal limiting distributions for systems of linear equations in random sets

Rué,

Wötzel

2021

Preprint

Self Cite

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show abstract

“…Their various properties were utilized in many other areas, both within and outside combinatorics. These include, among others, (hypergraph) Ramsey theory [4, 22, 25, 31-33, 49, 50], (hypergraph) Turán theory [1,2,36,37,39,40], other problems in extremal combinatorics, [6,11,28,41,44,45], number theory [35,43,47,48], geometry [19,38] and computer science [3,7,8,15].…”

Section: The Automorphism Groupmentioning

confidence: 99%

The automorphism group of projective norm graphs

Bayer

Mészáros

Rónyai

et al. 2023

AAECC

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The projective norm graphs are central objects to extremal combinatorics. They appear in a variety of contexts, most importantly they provide tight constructions for the Turán number of complete bipartite graphs $$K_{t,s}$$ K t , s with $$s>(t-1)!$$ s > ( t - 1 ) ! . In this note we deepen their understanding further by determining their automorphism group.

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“…Their various properties were utilized in many other areas, both within and outside combinatorics. These include, among others, (hypergraph) Ramsey theory [6,35,40,44,45,46,66,67], (hypergraph) Turán theory [7,3,48,49,52,53], other problems in extremal combinatorics, [2,13,42,55,58,59], number theory [50,57,64,65], geometry [27,51] and computer science [1,9,10,23].…”

Section: The Projective Norm Graphsmentioning

confidence: 99%

Exploring Projective Norm Graphs

Bayer¹,

Mészáros²,

Rónyai³

et al. 2019

Preprint

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The projective norm graphs NG(q, t) provide tight constructions for the Turán number of complete bipartite graphs K t,s with s > (t − 1)!. In this paper we determine their automorphism group and explore their small subgraphs. To this end we give quite precise estimates on the number of solutions of certain equation systems involving norms over finite fields. The determination of the largest integer s t , such that the projective norm graph NG(q, t) contains K t,st for all large enough prime powers q is an important open question with far-reaching general consequences. The best known bounds, t − 1 ≤ s t ≤ (t − 1)!, are far apart for t ≥ 4. Here we prove that NG(q, 4) does contain (many) K 4,6 for any prime power q not divisble by 2 or 3. This greatly extends recent work of Grosu, using a completely different approach. Along the way we also count the copies of any fixed 3-degenerate subgraph, and find that projective norm graphs are quasirandom with respect to this parameter. Some of these results also extend the work of Alon and Shikhelman on generalized Turán numbers. Finally we also give a new, more elementary proof for the K 4,7 -freeness of NG(q, 4).

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Counting configuration–free sets in groups

Cited by 3 publications

References 35 publications

Normal limiting distributions for systems of linear equations in random sets

Normal limiting distributions for systems of linear equations in random sets

The automorphism group of projective norm graphs

Exploring Projective Norm Graphs

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