Article
On the density of sets of divisors
Discrete Mathematics 01/1995; 137(s 1–3):345–349. DOI:10.1016/0012365X(93)E0114J
Source: DBLP

Article: A generalization of Sauer's lemma
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ABSTRACT: We generalize Sauer's lemma to multivalued functions, proving tight bounds on the cardinality of subsets of ∏i = 1m {0, …, Nm} which avoid certain patterns. In addition, we give an application of this result, bounding the uniform rate of convergence of empirical estimates of the expectations of a set of random variables to their true expectations.Journal of Combinatorial Theory, Series A. 01/1995;  [show abstract] [hide abstract]
ABSTRACT: This paper surveys various results concerning forbidden congurations that have been obtained by Aldred, Anstee, Barekat, Chervonenkis, Dunwoody, Farber, Ferguson, Fleming, Frankl, Furedi, Griggs, Gronau, Kamoosi, Karp, Keevash, Murty, Pach, Perles, Quinn, Ryan, Sali, Sauer, Shelah, and Vapnik to name a few. Let F be a k ' (0,1)matrix (the forbidden conguration). We dene a matrix to be simple if it is a (0,1)matrix with no repeated columns. Assume m is given and assume A is an m n simple matrix which has no submatrix which is a row and column permutation of F. We dene forb(m; F ) as the best possible upper bound on n depending on m and F. We seek exact values for forb(m; F ) as well as seeking asymptotic results for forb(m; F ) for a xed F and as m tends to innit y. A conjecture of Anstee and Sali predicts the asymptotically best constructions from which to derive the asymptotics of forb(m; F ).
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