Article

# On the density of sets of divisors

Department of Mathematics and Statistics, University of Otago, Taieri, Otago, New Zealand

Discrete Mathematics (Impact Factor: 0.56). 01/1995; 137(s 1–3):345–349. DOI: 10.1016/0012-365X(93)E0114-J Source: DBLP

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**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 ). -
##### 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. -
##### Article: Shattering News

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**ABSTRACT:**We explore the concepts of shattered and order-shattered sets. In particular, for every family ℱ of subsets of {1,2,…,m} there are exactly |ℱ| subsets of {1,2,…,m} order-shattered by ℱ. This provides proofs and strengthenings of the result of Sauer, Perles and Shelah, Vapnik and Chervonenkis (sometimes known as Sauer's lemma) and a new approach to the reverse Sauer Inequality of Bollobás and Radcliffe. We characterize those sets which can be order-shattered by a uniform family and those sets which can be order-shattered by an antichain. We also give an algebraic interpretation of order shattering using Gröbner bases. This results in sharpening of a theorem of Frankl and Pach. It also points out an interesting and promising connection between combinatorial and algebraic objects.