The trace of a family of sets A on a set X is A|X = {A ∩ X : A ∈ A}. If A is a family of k-sets from an n-set such that for any r-subset X the trace A|X does not contain a maximal chain, then how large can A be? Patkós conjectured that, for n sufficiently large, the size of A is at most ( n−k+r−1 r−1 ) . Our aim in this paper is to prove this conjecture.

We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most c, where c ≈ 1.6702. The best known lower bound for this number, due to Tomescu, is ad, where d = 10 1 5 ≈ 1.5849 and a is a constant.

Dynkin’s classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to matrix Lie superalgebras, i.e., the Lie subsuperalgebras of gl(p|q).

We show that for hypergraphs of bounded edge size, the problem of extending a given list of maximal independent sets is NC-reducible to the computation of an arbitrary maximal independent set for an induced sub-hypergraph. The latter problem is known to be in RNC. In particular, our reduction yields an incremental RNC dualization algorithm for hypergraphs of bounded edge size, a problem previou...

The independent domination number of a graph G, denoted i(G), is the minimum cardinality of a maximal independent set of G. A maximal independent set of cardinality i(G) in G we call an i(G)-set. In this paper we provide a constructive characterization of trees G that have two disjoint i(G)-sets. © 2005 Elsevier B.V. All rights reserved.