We establish a homomorphism of finite linear lattices onto the Boolean lattices via a group acting on linear lattices. By using this homomorphism we prove the intersecting antichains in finite linear lattices satisfy an LYM-type inequality, as conjectured by Erdos, Faigle and Kern, and we state a Kruskal-Katona type theorem for the linear lattices. (C) 2011 Elsevier Inc. All rights reserved.

Let G(X, Y) be a connected, non-complete bipartite graph with vertical bar X vertical bar <= vertical bar Y vertical bar. An independent set A of G(X, Y) is said to be trivial if A subset of X or A subset of Y. Otherwise, A is nontrivial. By alpha(X, Y) we denote the maximum size of nontrivial independent sets of G(X, Y). We prove that if the automorphism group of G(X, Y) is transitive and primitive on X and Y. respectively, then alpha(X, Y) = vertical bar Y vertical bar - d(X) + 1, where d(X) is the degree of vertices in X. We also give the structures of maximum-sized nontrivial independent sets of G(X, Y). Consequently, these results give the sizes and structures of maximum-sized cross-t-intersecting families of finite sets, finite vector spaces and permutations, as well as the sizes and structures of maximum-sized cross-Sperner families of finite sets and finite vector spaces. (C) 2012 Elsevier Inc. All rights reserved.

Let X be a finite set and p subset of 2(X) the power set of X, satisfying three conditions: (a) p is an ideal in 2(X) that is, if A is an element of p and B subset of A. then B is an element of p; (b) for A is an element of 2(X) with vertical bar A vertical bar >= 2, A is an element of p if {x, y} is an element of p for any x, y is an element of A with x not equal y; (c) {x} is an element of p for every x is an element of X. The pair (X, p) is called a symmetric system if there is a group P transitively acting on X and preserving the ideal p. A family {A1, A2,..., A(m)} subset of 2(X) is said to be a cross-p-family of X if {a, b} is an element of p for any a is an element of A(i) and b is an element of A(j) with i not equal j. We prove that if (X, p) is a symmetric system and {A1, A2,..., A(m)) subset of 2(X) is a cross-p-family of X, then Sigma(m)(i=1) vertical bar Ai vertical bar <= (vertical bar X vertical bar)(m alpha(X, p)) (if m >= vertical bar X vertical bar/alpha(X,p),) (if m <= vertical bar X vertical bar/alpha(X,p), m alpha(X,p)) where alpha(X, p) = max {vertical bar A vertical bar: A is an element of p}. This generalizes Hilton's theorem on cross-intersecting families of finite sets, and provides analogs for cross-t-intersecting families of finite sets, finite vector spaces and permutations, etc. Moreover, the primitivity of symmetric systems is introduced to characterize the optimal families. (C) 2010 Elsevier Inc. All rights reserved.