Constructing dense grid-free linear $3$-graphs

Gishboliner, Lior; Shapira, A.

doi:10.1090/proc/15673

Cited by 11 publications

(9 citation statements)

References 6 publications

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“…However, Füredi and Ruszinkó conjectured that similar asymptotics also hold for grid-free 3uniform hypergraphs. For significant progress on this, see [20].…”

Section: 2mentioning

confidence: 99%

Conflict-free hypergraph matchings

Glock¹,

Joos²,

Kim³

et al. 2022

Preprint

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A celebrated theorem of Pippenger, and Frankl and Rödl states that every almostregular, uniform hypergraph H with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a conflict-free matching, where conflicts are encoded via a collection C of subsets C ⊆ E(H). We say that a matching M ⊆ E(H) is conflict-free if M does not contain an element of C as a subset. Under natural assumptions on C, we prove that H has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called "high-girth" Steiner systems. Our main tool is a random greedy algorithm which we call the "conflict-free matching process".

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“…However, Füredi and Ruszinkó conjectured that similar asymptotics also hold for grid-free 3uniform hypergraphs. For significant progress on this, see [20].…”

Section: 2mentioning

confidence: 99%

Conflict-free hypergraph matchings

Glock¹,

Joos²,

Kim³

et al. 2022

Preprint

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“…was supported in part by an NSERC Discovery grant and OTKA K 119528 grant. The author is thankful to Lior Gishboliner for pointing out the crucial reference [19].…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…This conjecture was refuted by Gishboliner and Shapira giving a nice construction for 3-uniform hypergraphs on n vertices with ∼ n 2 16 edges without a tic-tac-toe [19]. Earlier, Füredi and Ruszinkó conjectured that there are 3-uniform hypergraphs on n vertices with ∼ n 2 6 edges (almost Steiner triple systems) without a tic-tac-toe [17].…”

Section: Introductionmentioning

confidence: 99%

On the structure of pointsets with many collinear triples

Solymosi¹

2022

Preprint

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It is conjectured that if a finite set of points in the plane contains many collinear triples then there is some structure in the set. We are going to show that under some combinatorial conditions such pointsets contain special configurations of triples, proving a case of Elekes' conjecture. Using the techniques applied in the proof we show a density version of Jamison's theorem. If the number of distinct directions between many pairs of points of a pointset in convex position is small, then many points are on a conic.

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“…Kim, Lee and Lee [17] proved Sidorenko's conjecture for grids (in arbitrary dimension). Füredi and Ruszinkó [13] studied the maximum number of hyperedges that an r-uniform linear hypergraph can have without containing a certain r × r hypergraph grid; see also [15].…”

Section: Introductionmentioning

confidence: 99%