Old and New Results for the Weighted t-Intersection Problem via AK-Methods

Bey, Christian; Engel, Konrad

doi:10.1007/978-1-4757-6048-4_5

Cited by 22 publications

(31 citation statements)

References 12 publications

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“…A sharp version of this result was first proved in [2], and can be deduced from the main result of [1]. See also [14], [24], [46] for subsequent related statements. Here we give a much simpler, self-contained proof of a somewhat weaker estimate that suffices for our purpose.…”

Section: Hypergraph Coloringmentioning

confidence: 74%

Paul Erdős and Probabilistic Reasoning

Alon

2013

Bolyai Society Mathematical Studies

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One of the major contributions of Paul Erdős is the development of the Probabilistic Method and its applications in Combinatorics, Graph Theory, Additive Number Theory and Combinatorial Geometry. This short paper describes some of the beautiful applications of the method, focusing on the long-term impact of the work, questions and results of Erdős. This is mostly a survey, but it contains a few novel results as well. The Probabilistic MethodThe Probabilistic Method is one of the most significant contributions of Paul Erdős, and part of his greatness is the fact that applications of the probabilistic method and of random graphs have become so common that it is now possible to use those without explicitly mentioning him. The method is a powerful tool with numerous applications in Combinatorics, Graph theory, Additive Number Theory and Geometry and had an immense impact on the development of theoretical Computer Science as well. The results and tools are far too numerous to cover in a short survey, even if the focus is only on those influenced directly by the work and problems of Erdős, and thus this paper is mainly a selection of topics that illustrate the method, and is not meant to be a comprehensive treatment of the whole area. Several books that contain more material on the subject are [13], [18], [54], [61].It is convenient to classify the applications of probabilistic techniques in Discrete Mathematics into three groups. The first one deals with the study of random combinatorial objects, like random graphs or random matrices. The results here are essentially results in Probability Theory, although many of them are motivated by problems in Combinatorics. The second group consists of probabilistic constructions. These are applications of probabilistic arguments in order to prove the existence of combinatorial structures which satisfy a list of prescribed properties. Existence proofs of this type often supply extremal examples to various questions in Discrete Mathematics. The third group, which contains some of the most striking examples, focuses on the application of probabilistic reasoning in the proofs of deterministic statements whose formulation does not give any indication that randomness may be helpful in their study.

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Section: Hypergraph Coloringmentioning

confidence: 74%

Paul Erdős and Probabilistic Reasoning

Alon

2013

Bolyai Society Mathematical Studies

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“…Let ν be the weight vector given by (3), and let r ∈ [0, k − t + 1] be determined as in Theorem 3 ′ . Then there exists an A ∈ I(n, t) with ν(A) = M(n, t, ν) which is t-intersecting in [t + 2r].…”

Section: Bey / Discrete Applied Mathematics ( ) -mentioning

confidence: 99%

An intersection theorem for systems of finite sets

Bey

2017

Discrete Applied Mathematics

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“…They gave two proofs [1,3], and both of them are purely combinatorial and based on shifting and clever exchange operations. For their methods we recommend an excellent survey by Bey and Engel [10].…”

Section: ⌋mentioning

confidence: 99%

Invitation to intersection problems for finite sets

Frankl

Tokushige

2016

Journal of Combinatorial Theory, Series A

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Abstract. Extremal set theory is dealing with families, F of subsets of an nelement set. The usual problem is to determine or estimate the maximum possible size of F, supposing that F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any r subsets in F have at least t elements in common, all the sizes of pairwise intersections belong to a fixed set, L of natural numbers, there are no s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bounds.

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Old and New Results for the Weighted t-Intersection Problem via AK-Methods

Cited by 22 publications

References 12 publications

Paul Erdős and Probabilistic Reasoning

Paul Erdős and Probabilistic Reasoning

An intersection theorem for systems of finite sets

Invitation to intersection problems for finite sets

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