Counterexample to the Frankl-Pach conjecture for uniform, dense families

Ahlswede, Rudolf; Khachatrian, Levon H.

doi:10.1007/bf01200912

Cited by 7 publications

(35 citation statements)

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“…We combine these two results to prove Theorem 1.4. Note that we have no intermediate asymptotic values for forb(m, F ) such as Θ(m k− 1 2 ) for k-rowed F . Theorem 1.4 was a prediction of the Conjecture in [6].…”

Section: Boundary Between O(m K−1 ) and θ(M K )mentioning

confidence: 99%

“…So there is no copy of 2 · K k on these rows. Nor can we find even K k on a submatrix involving two rows from a single factor, as = 0, 1,k − 1,k implies that on every pair of rows of K k there is both 1 1 and 0 0 , which are not configurations in I m/k and I m/k , respectively. Therefore 2 · K k is not a configuration in A.…”

Section: Boundary Between O(m K−1 ) and θ(M K )mentioning

confidence: 99%

“…We may assume that F does not contain 2 · 0 = 2 · K 0 k . Therefore, as F is not a configuration in F k (t), F contains [2 · I k |G], where G is some matrix with 1 1 in every pair of rows. Similarly to before the k-fold product A consisting of one factor I m/k and k − 1 factors I m/k witnesses that forb(m, F ) is Θ(m k ).…”

Section: Boundary Between O(m K−1 ) and θ(M K )mentioning

confidence: 99%

“…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: 98%

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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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Section: Boundary Between O(m K−1 ) and θ(M K )mentioning

confidence: 99%

Section: Boundary Between O(m K−1 ) and θ(M K )mentioning

confidence: 99%

Section: Boundary Between O(m K−1 ) and θ(M K )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Linear algebra methods for forbidden configurations

Anstee

Fleming

2011

Combinatorica

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“…Therefore the conjecture of Frankl and Pach, if true, generalizes the wellknown Erdős-Ko-Rado Theorem [5]. However, Ahlswede and Khachatrian [1] disproved it by constructing a G ⊆ [n] k of size n−1 k−1 + n−4 k−3 that contains no shattered k-set when k ≥ 3 and n ≥ 2k. Combining this with the upper bound in [7], for k ≥ 3 and n ≥ 2k,…”

Section: Introductionmentioning

confidence: 99%

On the VC-dimension of uniform hypergraphs

Mubayi

Zhao

2006

J Algebr Comb

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The VC-Dimension of Sperner Systems

Greco

1999

Journal of Combinatorial Theory, Series A

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A family F of subsets of a finite set X shatters a set D X, if the intersections of the members of F with D coincide with the power set of D. The maximum size of a set shattered by F is the VC-dimension (or density) of the system. P. Frankl , is a Sperner system if for every two members of the system none contains the other (i.e., F is an antichain in the boolean lattice). A chain is a sequence of sets A i [m], 0 i n such thatA chain is complete if n=m and |A i | =i for every 0 i m.If D is a subset of [m] the trace of F on D is

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Counterexample to the Frankl-Pach conjecture for uniform, dense families

Cited by 7 publications

References 5 publications

Linear algebra methods for forbidden configurations

Linear algebra methods for forbidden configurations

On the VC-dimension of uniform hypergraphs

The VC-Dimension of Sperner Systems

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