Article

On the density of sets of divisors

Department of Mathematics and Statistics, University of Otago, Taieri, Otago, New Zealand
Discrete Mathematics (Impact Factor: 0.57). 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.

0 Followers
 · 
15 Views
  • [Show abstract] [Hide abstract]
    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. DOI:10.1023/A:1022934815185 · 0.72 Impact Factor
  • [Show abstract] [Hide abstract]
    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.
    Graphs and Combinatorics 02/2002; 18(1):59-73. DOI:10.1007/s003730200003 · 0.33 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 ).

Preview

Download
0 Downloads
Available from