On the VC-dimension of uniform hypergraphs

Mubayi, Dhruv; Zhao, Yi

doi:10.1007/s10801-006-0025-4

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Cited by 9 publications

(6 citation statements)

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“…6. For a recent partial improvement of the Frankl-Pach bound we refer to Mubayi and Zhao [20]. Shattering and related notions have many important applications in mathematics and computer science.…”

Section: Discussionmentioning

confidence: 99%

Multivalued Generalizations of the Frankl–pach Theorem

Hegedüs

Rónyai

2012

J. Algebra Appl.

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Section: Discussionmentioning

confidence: 99%

Multivalued Generalizations of the Frankl–pach Theorem

Hegedüs

Rónyai

2012

J. Algebra Appl.

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“…In the Set Theory context a configuration is called a trace. A survey of trace results is in [11] and, for example, a variety of results have been obtained for traces of uniform set systems [1,9,[13][14][15]. There are interesting results when forbidding more than one configuration such as the striking result of Balogh and Bollobás [7] and related results [8].…”

Section: Introductionmentioning

confidence: 99%

Linear algebra methods for forbidden configurations

Anstee

Fleming

2011

Combinatorica

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We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m, F ) as the maximum number of columns in a simple m-rowed matrix A for which no k × l submatrix of A is a row and column permutation of F . In set theory notation, F is a forbidden trace. For all k-rowed F (simple or nonsimple) Füredi has shown that forb(m, F ) is O(m k ). We are able to determine for which k-rowed F we have that forb(m, F ) is O(m k−1 ) and for which k-rowed F we have that forb(m, F ) is Θ(m k ).We need a bound for a particular choice of F . Define D12 to be the k ×(2 k −2 k−2 −1) (0,1)-matrix consisting of all nonzero columns on k rows that do not haveˆ1 1˜i n rows 1 and 2. Let 0 denote the column of k 0's. Define F k (t) to be the concatenation of 0 with t + 1 copies of D12. We are able to show that forb(m, F k (t)) is Θ(m k−1 ). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Füredi and Sali. We provide a novel application of these methods.The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed F of forb(m, F ).

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“…They proved that if A ⊂ [n] (k) is a family with more than n k−1 sets, then there is a k-subset X of [n] such that A| X = P(X). This bound is not sharp and was improved later by Mubayi and Zhao [3], but the exact bound is still unknown.…”

Section: Introductionmentioning

confidence: 99%