Geometric constructions for Ramsey-Turán theory

Liu, Hong; Reiher, Christian; Sharifzadeh, Maryam; Staden, Katherine

doi:10.48550/arxiv.2103.10423

Cited by 4 publications

(4 citation statements)

References 32 publications

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“…This turned out to be sharp as Bollobás and Erdős [7] provided a matching lower bound using an ingenious geometric construction. There are some recent exciting developments in this area [3,4,16,24,31,33]. For further information on Ramsey-Turán theory the reader is referred to a comprehensive survey [37] by Simonovits and Sós.…”

Section: Motivationmentioning

confidence: 99%

Clique-factors in graphs with sublinear -independence number

Han

Wang

et al. 2023

Combinator. Probab. Comp.

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Given a graph $G$ and an integer $\ell \ge 2$ , we denote by $\alpha _{\ell }(G)$ the maximum size of a $K_{\ell }$ -free subset of vertices in $V(G)$ . A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$ -vertex graphs $G$ with $\alpha _{\ell }(G) = o(n)$ , which can be seen as a Ramsey–Turán variant of the celebrated Hajnal–Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$ -factors in $n$ -vertex graphs $G$ with $\alpha _\ell (G)=n^{1-o(1)}$ for all $r\ge \ell \ge 2$ .

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Section: Motivationmentioning

confidence: 99%

Clique-factors in graphs with sublinear -independence number

Han

Wang

et al. 2023

Combinator. Probab. Comp.

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“…This turned out to be sharp as Bollobás and Erdős [7] provided a matching lower bound using an ingenious geometric construction. There are some recent exciting developments in this area [4,5,16,24,30,32]. For further information on Ramsey-Turán theory the reader is referred to a comprehensive survey [36] by Simonovits and Sós.…”

Section: Motivationmentioning

confidence: 99%

Clique-factors in graphs with sublinear $\ell$-independence number

Han¹,

Hu²,

Wang³

et al. 2022

Preprint

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Given a graph G and an integer ≥ 2, we denote by α (G) the maximum size of a K -free subset of vertices in V (G). A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in n-vertex graphs G with α (G) = o(n), which can be seen as a Ramsey-Turán variant of the celebrated Hajnal-Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for n-vertex graphs G with α (G) ∈ (n 1−γ , n 1−o(1) ) for any positive constant γ < −1 2 +2 .

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“…This turned out to be sharp as four years later Bollobás and Erdős [6] provided a matching lower bound using an ingenious geometric construction. There are some recent exciting developments in this area [2,3,33]. For further information on Ramsey-Turán theory the reader is referred to the comprehensive survey [42] by Simonovits and Sós.…”

Section: Ramsey-turán Theorymentioning

confidence: 99%

Embedding clique-factors in graphs with low $\ell$-independence number

Chang

Han

Kim³

et al. 2021

Preprint

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The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers , r and n with n ∈ rN, is it true that every n-vertex graph G with δ(G) ≥ max{ 1 2 , r− r }n + o(n) and α (G) = o(n) contains a K r -factor? We give a negative answer for the case when ≥ 3r 4 by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers r, with r > ≥ 3 4 r and µ > 0, there exist α > 0 and N such that for every n ∈ rN with n > N , every n-vertex graph G with δ(G) ≥ 1 2− (r−1) + µ n and α (G) ≤ αn contains a K r -factor. Here (r − 1) is the Ramsey-Turán density for K r−1 under the -independence number condition.

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Geometric constructions for Ramsey-Turán theory

Cited by 4 publications

References 32 publications

Clique-factors in graphs with sublinear -independence number

Clique-factors in graphs with sublinear -independence number

Clique-factors in graphs with sublinear $\ell$-independence number

Embedding clique-factors in graphs with low $\ell$-independence number

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