Hypergraphs with no tight cycles

Letzter, Shoham

doi:10.48550/arxiv.2106.12082

Cited by 3 publications

(7 citation statements)

References 11 publications

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“…Given a 3-graph G, let p(G) denote the number of 1-dimensional faces of G, that is, the number of pairs of vertices which appear in an edge. As proved by Letzter [33], slightly improving the result of Sudakov and Tomon [42], if a 3-graph G satisfies e(G) = Ω(p(G)(log p(G)) 5 ), then G contains a tight cycle. A similar strengthening of Theorem 1.5 also holds.…”

Section: Cycles In Simplicial Complexesmentioning

confidence: 82%

“…This was disproved in a strong sense by B. Janzer [17], who showed that an r-graph on n vertices can have Ω(n r−1 log n/ log log n) edges without containing a tight cycle. On the other hand, Sudakov and Tomon [42] proved the upper bound n r−1+o (1) , which was further improved by Letzter [33] to O(n r−1 (log n) 5 ). However, none of these proofs seem to extend to give any bounds on the extremal numbers of tight cycle of fixed length ℓ. Verstraëte [43] proposed the conjecture that if r divides ℓ, then any r-graph with n vertices containing no tight cycle of length ℓ can have at most O(n r−1+(r−1)/ℓ ) edges.…”

Section: Cycles In Simplicial Complexesmentioning

confidence: 96%

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Robust (rainbow) subdivisions and simplicial cycles

Tomon¹

2022

Preprint

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We present several results in extremal graph and hypergraph theory of topological nature. First, we show that if α > 0 and ℓ = Ω( 1 α log 1 α ) is an odd integer, then every graph G with n vertices and at least n 1+α edges contains an ℓ-subdivision of the complete graph K t , where t = n Θ(α) . Also, this remains true if in addition the edges of G are properly colored, and one wants to find a rainbow copy of such a subdivision. In the sparser regime, we show that properly edge colored graphs on n vertices with average degree (log n) 2+o(1) contain rainbow cycles, while average degree (log n) 6+o(1) guarantees rainbow subdivisions of K t for any fixed t, thus improving recent results of Janzer and Jiang et al., respectively. Furthermore, we consider certain topological notions of cycles in pure simplicial complexes (uniform hypergraphs). We show that if G is a 2-dimensional pure simplicial complex (3-graph) with n 1-dimensional and at least n 1+α 2-dimensional faces, then G contains a triangulation of the cylinder and the Möbius strip with O( 1 α log 1 α ) vertices. We present generalizations of this for higher dimensional pure simplicial complexes as well.In order to prove these results, we consider certain (properly edge colored) graphs and hypergraphs G with strong expansion. We argue that if one randomly samples the vertices (and colors) of G with not too small probability, then many pairs of vertices are connected by a short path whose vertices (and colors) are from the sampled set, with high probability.

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Section: Cycles In Simplicial Complexesmentioning

confidence: 82%

Section: Cycles In Simplicial Complexesmentioning

confidence: 96%

Robust (rainbow) subdivisions and simplicial cycles

Tomon¹

2022

Preprint

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“…To show that there is a short rainbow x, y-path in G, we first apply tools due to Jiang, Methuku and Yepremyan [5] and Letzter [8] to show that there is a set of vertices U of size Ω(n) such that for each v ∈ U there is such a short rainbow x, v-path P (v) and a short rainbow y, v-path Q(v), both of which avoid C, such that no colour is used on too many of these paths P (v) and Q(v). It easily follows that for almost all pairs (u, v) with u, v ∈ U , the paths P (u) and Q(v) are colour-disjoint.…”

Section: Overview Of the Proofsmentioning

confidence: 99%

“…We also prove a strengthening of Theorem 1.2, regarding 'rooted' rainbow subdivisions of K m in expanders (see Theorem 6.1). For this stronger version, in addition to the ingredients used for proving Theorem 1.2, we use the framework of [5] and an additional idea used by Letzter in [8] (see Lemma 3.7).…”

Section: Introductionmentioning

confidence: 99%

Rainbow clique subdivisions and blow-ups

Jiang¹,

Letzter²,

Methuku³

et al. 2021

Preprint

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We show that for every integer m ≥ 2 and large n, every properly edge-coloured graph on n vertices with at least n(log n) 60 edges contains a rainbow subdivision of K m .Using the same framework, we also prove a result about the extremal number of r-blow-ups of subdivisions. We show that for integers r, m ≥ 2 and large n, every graph on n vertices with at least n 2− 1 r (log n) 60 r edges has an r-blow-up of a subdivision of K m .Both results are sharp up to a polylogarithmic factor. Our proofs use the connection between mixing time of random walks and expansion in graphs.

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“…Turán type problems for tight cycles have a long history, dating back to an old question of Sós (see [21]), and appear in relation to seemingly unrelated topics [4]. We refer the reader particularly to the report [21] from the 2011 American Institute of Mathematics (AIM) workshop "Hypergraph Turán Problem", which contains many such problems, and to [15,16,19,34] for some recent specific results. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Uniform Turán density of cycles

Bucić¹,

Cooper²,

Kráľ³

et al. 2021

Preprint

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In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K (3)− 4 and K (3) 4 . The former question was solved only recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 97 (2018), 77-97], while the latter still remains open for almost 40 years. In addition to K (3)− 4 , the only 3-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), 77-97] and a specific family with uniform Turán density equal to 1/27.We develop new tools for embedding hypergraphs in host hypergraphs with positive uniform density and apply them to completely determine the uniform Turán density of a fundamental family of 3uniform hypergraphs, namely tight cycles C(3) ℓ . The uniform Turán density of C(3) ℓ , ℓ ≥ 5, is equal to 4/27 if ℓ is not divisible by three, and is equal to zero otherwise. The case ℓ = 5 resolves a problem suggested by Reiher.

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Hypergraphs with no tight cycles

Cited by 3 publications

References 11 publications

Robust (rainbow) subdivisions and simplicial cycles

Robust (rainbow) subdivisions and simplicial cycles

Rainbow clique subdivisions and blow-ups

Uniform Turán density of cycles

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