Article

# 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

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**ABSTRACT:**We describe (reduced) Grbner bases of the ideal of polynomials over a field, which vanish on the set of characterisic vectors of the complete unifom families (\text d[n] )(_{{\text{ }}d}^{[n]} ) . An interesting feature of the results is that they are largely independent of the monomial order selected. The bases depend only on the ordering of the variables. We can thus use past results related to the lex order in the presence of degree-compatible orders, such as deglex. As applications, we give simple proofs of some known results on incidence matrices.Journal of Algebraic Combinatorics 02/2003; 17(2):171-180. · 0.63 Impact Factor - [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 ). -
##### 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;

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