Maximum union-free subfamilies

Fox, Jacob; Lee, Choongbum; Sudakov, Benny

doi:10.1007/s11856-012-0017-0

Cited by 4 publications

(6 citation statements)

References 14 publications

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“…Since the first version of this manuscript, Fox, Lee, and Sudakov [3] verified the present authors' conjecture (see later as Problem 5) and proved a matching lower bound showing that f (m, a-union free) ≥ max{a,…”

supporting

confidence: 68%

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Large $B_d$-Free and Union-free Subfamilies

Barát¹,

Füredi²,

Kantor³

et al. 2012

SIAM J. Discrete Math.

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supporting

confidence: 68%

“…Therefore |G ℓ | ≤ 2k − 1 by a slight strenghtening of the result of Erdős and Shelah (see [3]). Putting these observations together, using |G| = |G ℓ | and t ≥ 1, we obtain (4).…”

Section: Union-free Subfamiliesmentioning

confidence: 67%

Large $B_d$-Free and Union-free Subfamilies

Barát¹,

Füredi²,

Kantor³

et al. 2012

SIAM J. Discrete Math.

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“…They called this family B 2 , proved that f (m, B 2 ) ≤ (3/2)m 2/3 and conjectured that this bound is asymptotically tight. This conjecture was settled by Barát, Füredi, Kantor, Kim and Patkós in 2012 [5], who also considered more general problems (see [4] for further work).…”

Section: Introductionmentioning

confidence: 95%

“…This function was introduced by Erdős and Komlós in 1969 [1], who considered the case when H is the (infinite) family of hypergraphs A, B, C with A ∪ B = C. The problem was further studied by Kleitman [2], and later by Erdős and Shelah [3], and finally settled by Fox, Lee and Sudakov [4] who proved that f (m, H) = √ 4m + 1 − 1.…”

Section: Introductionmentioning

confidence: 99%

Maximum $\mathcal H$-free subgraphs

Mubayi,

Mukherjee

2019

Preprint

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Given a family of hypergraphs H, let f (m, H) denote the largest size of an H-free subgraph that one is guaranteed to find in every hypergraph with m edges. This function was first introduced by Erdős and Komlós in 1969 in the context of union-free families, and various other special cases have been extensively studied since then. In an attempt to develop a general theory for these questions, we consider the following basic issue: which sequences of hypergraph families {Hm} have bounded f (m, Hm) as m → ∞? A variety of bounds for f (m, Hm) are obtained which answer this question in some cases. Obtaining a complete description of sequences {Hm} for which f (m, Hm) is bounded seems hopeless.

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“…As above, a natural spanoid S F will have the inference S → i whenever the set F i contains the common intersection of all F j : j ∈ S. Such situations (and hence, spanoids) arise in many questions of extremal set theory, for example the study of (weak) sunflowers, or families with certain forbidden intersection (or union) patterns, e.g. [FLS12,EFF85,F 96].…”

Section: Intersecting Set Systemsmentioning

confidence: 99%

Spanoids - an abstraction of spanning structures, and a barrier for LCCs

Dvir¹,

Gopi²,

Gu³

et al. 2018

Preprint

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We introduce a simple logical inference structure we call a spanoid (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry (point-line incidences), algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation), network theory (gossip / infection processes) and coding theory. We initiate a thorough investigation of spanoids, from computational and structural viewpoints, focusing on parameters relevant to the applications areas above and, in particular, to questions regarding Locally Correctable Codes (LCCs).One central parameter we study is the rank of a spanoid, extending the rank of a matroid and related to the dimension of codes. This leads to one main application of our work, establishing the first known barrier to improving the nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. On the one hand, we prove that the KT bound (and its more recent refinements) holds for the much more general setting of spanoid rank. On the other hand we show that there exist (random) spanoids whose rank matches these bounds. Thus, to significantly improve the known bounds one must step out of the spanoid framework.Another parameter we explore is the functional rank of a spanoid, which captures the possibility of turning a given spanoid into an actual code. The question of the relationship between rank and functional rank is one of the main questions we raise as it may reveal new avenues for constructing new LCCs (perhaps even matching the KT bound). As a first step, we develop an entropy relaxation of functional rank to create a small constant gap and amplify it by tensoring to construct a spanoid whose functional rank is smaller than rank by a polynomial factor. This is evidence that the entropy method we develop can prove polynomially better bounds than KT-type methods on the dimension of LCCs.To facilitate the above results we also develop some basic structural results on spanoids including an equivalent formulation of spanoids as set systems and properties of spanoid products. We feel that given these initial findings and their motivations, the abstract study of spanoids merits further investigation. We leave plenty of concrete open problems and directions.

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Maximum union-free subfamilies

Cited by 4 publications

References 14 publications

Large $B_d$-Free and Union-free Subfamilies

Large $B_d$-Free and Union-free Subfamilies

Maximum $\mathcal H$-free subgraphs

Spanoids - an abstraction of spanning structures, and a barrier for LCCs

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