On the density of sets of divisors

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

ABSTRACT Consider the lattice of divisors of n, [1, n]. For any downset (ideal) ℐ in [1, n] we get a forbidden configuration theorem of the type that if a set of divisors D avoids certain configurations, then |D|⩾|ℐ|. If we let ℓ be the set of minimal elements of [1, n] not in ℐ, then we forbid in D the configurations C(s) (defined in the paper) for s∈ℒ. This generalizes a result of Alon and in turn generalizes a result of Sauer, Perles and Shelah.

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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;
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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 ).


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