Short proofs of three results about intersecting systems

Balogh, József; Linz, William

doi:10.48550/arxiv.2104.00778

Cited by 3 publications

(6 citation statements)

References 13 publications

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“…Finally in the last section we discuss possible extensions to non-trivial r-wise intersecting families for r ≥ 4, and a related k-uniform problem. In particular we include counterexamples to recent conjectures posed by O'Neill and Versträete [15], and by Balogh and Linz [3].…”

Section: Introductionmentioning

confidence: 92%

“…Theorem 3 suggests that F 3 (n, k, 1) could be a counterexample to the conjecture if n and k are sufficiently large and roughly Balogh and Linz [3] made another conjecture. They constructed a family BL r (n, k) as possibly the largest non-trivial r-wise intersecting families in [n] k .…”

Section: Subcasementioning

confidence: 99%

“…For simplicity we often write G I or x I without braces and commas, e.g., we write G 12 to mean G {1,2} . Let Ī denote [3]…”

Section: Proof Of Theoremmentioning

confidence: 99%

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The maximum measure of non-trivial 3-wise intersecting families

Tokushige¹

2022

Preprint

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Let G be a family of subsets of an n-element set. The family G is called nontrivial 3-wise intersecting if the intersection of any three subsets in G is non-empty, but the intersection of all subsets is empty. For a real number p ∈ (0, 1) we define the measure of the family by the sum of p |G| (1 − p) n−|G| over all G ∈ G. We determine the maximum measure of non-trivial 3-wise intersecting families. We also discuss the uniqueness and stability of the corresponding optimal structure. These results are obtained by solving linear programming problems.

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Section: Introductionmentioning

confidence: 92%

Section: Subcasementioning

confidence: 99%

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The maximum measure of non-trivial 3-wise intersecting families

Tokushige¹

2022

Preprint

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“…A further study direction is to reduce the value of n 1 (k, d). Balogh and Linz [4], recently, gave a very short proof to greatly improve the lower bound for n in Corollary 1.6(1) by using the simple fact that every non-trivial d-wise intersecting family is (d − 1)intersecting. Since the largest non-trivial (d − 1)-intersecting families have been investigated systematically in [1,2,3] and they are exactly d-wise intersecting, as a simple corollary, Balogh and Linz showed that the largest non-trivial d-wise intersecting k-uniform family is isomorphic to one of H(k, d, 2) and H(k, d, k − d + 2) for any n > (1 + d 2 )(k − d + 2).…”

Section: Comparison Of the Sizes Of Extremal Familiesmentioning

confidence: 99%

The structure of maximal non-trivial d-wise intersecting uniform families with large sizes

Zhang¹,

Feng²

2022

Preprint

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For a positive integer d 2, a family F ⊆ [n] k is said to be d-wise intersecting ifWe provide a refinement of O'Neill and Verstraëte's Theorem about the structure of the largest and the second largest maximal non-trivial d-wise intersecting k-uniform families. We also determine the structure of the third largest and the fourth largest maximal non-trivial d-wise intersecting k-uniform families for any k > d + 1 4, and the fifth largest and the sixth largest maximal non-trivial 3-wise intersecting k-uniform families for any k 5, in the asymptotic sense. Our proofs are applications of the ∆-system method.

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“…Conjecture 4.6 would imply that it is the case when H = K t for all t ≥ 5. In 2012, the authors of [29] raised the question of whether the Erdős-Ko-Rado property holds whenever H is 2-connected, but a construction of Balogh and Linz [6] shows that when H = K s,t and t > 2 2s − 2s − 1, the Erdős-Ko-Rado property does not hold for the H-intersection problem, so even the condition of H being s-connected is insufficient to guarantee the Erdős-Ko-Rado property (for any s ∈ N).…”

Section: Imposing Extra Structure On the Ground Setmentioning

confidence: 99%

Intersection Problems in Extremal Combinatorics: Theorems, Techniques and Questions Old and New

Ellis¹

2021

Preprint

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The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul Erdős, Chao Ko and Richard Rado proved the (first) 'Erdős-Ko-Rado theorem' on the maximum possible size of an intersecting family of k-element subsets of a finite set. Since then, a plethora of results of a similar flavour have been proved, for a range of different mathematical structures, using a wide variety of different methods. Structures studied in this context have included families of vector subspaces, families of graphs, subsets of finite groups with given group actions, and of course uniform hypergraphs with stronger or weaker intersection conditions imposed. The methods used have included purely combinatorial methods such as shifting/compressions, algebraic methods (including linear-algebraic, Fourier analytic and representation-theoretic), and more recently, analytic, probabilistic and regularity-type methods. As well as being natural problems in their own right, intersection problems have revealed connections with many other parts of Combinatorics and with Theoretical Computer Science (and indeed with many other parts of Mathematics), both through the results themselves, and the methods used.In this survey paper, we discuss both old and new results (and both old and new methods), in the field of intersection problems. Many interesting open problems remain; we will discuss several such. For expositional and pedagogical purposes, we also take this opportunity to give slightly streamlined versions of proofs (due to others) of several classical results in the area. This survey is intended to be useful to graduate students, as well as to more established researchers. It is a somewhat personal perspective on the field of intersection problems, and is not intended to be exhaustive; the author apologises for any omissions.

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Short proofs of three results about intersecting systems

Cited by 3 publications

References 13 publications

The maximum measure of non-trivial 3-wise intersecting families

The maximum measure of non-trivial 3-wise intersecting families

The structure of maximal non-trivial d-wise intersecting uniform families with large sizes

Intersection Problems in Extremal Combinatorics: Theorems, Techniques and Questions Old and New

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