Grobner bases for complete l-wide families

Let n > 0, k, l be integers with 0 <= l - 1 <= k <= n, and consider the complete t-wide family F-k,F-l = {F subset of [n] : k-l < vertical bar F vertical bar <= k}. We describe (reduced) Grobner bases of the ideal of polynomials, over an arbitrary field F, which vanish on the characteristic vectors of the elements of F-k,F-l As an application, we obtain results on certain inclusion matrices related to F-k,F-l. We show that if 0 <= m <= min(k, n - k + l - 1) then [GRAPHICS] where F is an arbitrary field. We prove also a special case of a conjecture of Frankl related to the determination of the maximum number of subsets of [n] with no shattered set of size t and with no chain of size l + 1. The paper extends the results obtained for the case of uniform families (the case l = 1) in [11].