Hypergraphs with Minimum Positive Uniform Turán Density

Garbe, Frederik; Kráľ, Daniel; Lamaison, Ander

doi:10.1007/978-3-030-83823-2_24

Cited by 2 publications

(6 citation statements)

References 17 publications

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“…The first lemma follows from Lemma 3.3 by applying it to the reverse of reduced 3-graph A given in the statement of Lemma 3.3. We note that it also appears in [3,12]. Lemma 6.5 ([3, Lemma 4.4] or [12,Lemma 8]).…”

Section: Applications Of Theorem 13mentioning

confidence: 88%

“…We follow some notion introduced in Reiher's survey [20,Section 3]. It should be noted that the idea of reduced 3-graphs has also been mentioned in other related works [23,21,3,12,4]. Definition 2.1 (Reduced 3-graph).…”

Section: Reduced Hypergraphsmentioning

confidence: 99%

“…The next two lemmas are intended to identify representative vertices based on candidate sets from the top vertices and the right vertices of the constituents, respectively. It should be noted that the proofs of these two lemmas have been implicitly presented in [21] using an iterative approach, while alternative proofs can also be found in [12,3].…”

Section: Intersection Lemmasmentioning

confidence: 99%

“…Meanwhile, they [22] also proposed the following problem. Very recently, Bucić et al [3] determined the uniform Turán density of all tight 3-uniform cycles of length at least five; Garbe, Král' and Lamaison [12] shown a specific family of 3-graphs F with π (F ) = 1/27; Chen and Schülke [4] proved π (F ⋆ 5 ) = 1/4, where F ⋆ 5 is a 3-graph on 5 vertices that is obtained from K . Given a 3-graph F , the t-blow-up of F , denoted by F (t), is the 3-graph obtained from F by replacing every vertex of F by t copies of itself.…”

Section: Introductionmentioning

confidence: 99%

“…[3, Lemma 4.3] or[12, Lemma 10]). For any δ > 0 and n ∈ N, there exists N ∈ N such that the following holds.…”

mentioning

confidence: 99%

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Key events in cancer: Dysregulation of SREBPs

Zhao

et al. 2023

Front. Pharmacol.

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Lipid metabolism reprogramming is an important hallmark of tumor progression. Cancer cells require high levels of lipid synthesis and uptake not only to support their continued replication, invasion, metastasis, and survival but also to participate in the formation of biological membranes and signaling molecules. Sterol regulatory element binding proteins (SREBPs) are core transcription factors that control lipid metabolism and the expression of important genes for lipid synthesis and uptake. A growing number of studies have shown that SREBPs are significantly upregulated in human cancers and serve as intermediaries providing a mechanistic link between lipid metabolism reprogramming and malignancy. Different subcellular localizations, including endoplasmic reticulum, Golgi, and nucleus, play an indispensable role in regulating the cleavage maturation and activity of SREBPs. In this review, we focus on the relationship between aberrant regulation of SREBPs activity in three organelles and tumor progression. Because blocking the regulation of lipid synthesis by SREBPs has gradually become an important part of tumor therapy, this review also summarizes and analyzes several current mainstream strategies.

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Section: Applications Of Theorem 13mentioning

confidence: 88%

Section: Reduced Hypergraphsmentioning

confidence: 99%

Section: Intersection Lemmasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[3, Lemma 4.3] or[12, Lemma 10]). For any δ > 0 and n ∈ N, there exists N ∈ N such that the following holds.…”

mentioning

confidence: 99%

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Key events in cancer: Dysregulation of SREBPs

Zhao

et al. 2023

Front. Pharmacol.

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Uniform Turán density of cycles

Bucić

Cooper

Kráľ

et al. 2023

Trans. Amer. Math. Soc.

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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 H is the infimum over all d d for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least d d contains H H . In particular, they raise the questions of determining the uniform Turán densities of K 4 ( 3 ) − K_4^{(3)-} and K 4 ( 3 ) K_4^{(3)} . The former question was solved only recently by Glebov, Král’, and Volec [Israel J. Math. 211 (2016), pp. 349–366] and Reiher, Rödl, and Schacht [J. Eur. Math. Soc. 20 (2018), pp. 1139–1159], while the latter still remains open for almost 40 years. In addition to K 4 ( 3 ) − K_4^{(3)-} , the only 3 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), pp. 77–97] and a specific family with uniform Turán density equal to 1 / 27 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 3 3 -uniform hypergraphs, namely tight cycles C ℓ ( 3 ) C_\ell ^{(3)} . The uniform Turán density of C ℓ ( 3 ) C_\ell ^{(3)} , ℓ ≥ 5 \ell \ge 5 , is equal to 4 / 27 4/27 if ℓ \ell is not divisible by three, and is equal to zero otherwise. The case ℓ = 5 \ell =5 resolves a problem suggested by Reiher.

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Hypergraphs with Minimum Positive Uniform Turán Density

Cited by 2 publications

References 17 publications

Key events in cancer: Dysregulation of SREBPs

Key events in cancer: Dysregulation of SREBPs

Uniform Turán density of cycles

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