Let n,k,ℓ,t be positive integers with k≥ℓ≥t+2 and let X=X1⊎X2⊎⋯⊎Xk, |Xi|=n. A family F of ℓ-subsets of X is called a separated family if |F∩Xi|≤1 for all F∈F and i=1,2,…,k. A separated family F is called non-trivial t-intersecting if |F∩F′|≥t for all F,F′∈F and |∩{F:F∈F}| (t+1)(ℓ−t−1)2(k−t−1)+1.

Abstract. A graph [Formula: see text] is called [Formula: see text]-free if it does not contain any cycle of length [Formula: see text]. In 1962, Erdös (together with Gallai), and independently Andrásfai, proved that every [Formula: see text]-vertex triangle-free graph with more than [Formula: see text] edges is bipartite. In this paper, we extend their result and show that for [Formula: see text] and [Formula: see text], every [Formula: see text]-vertex [Formula: see text]-free graph with more than [Formula: see text] edges can be made bipartite by either deleting at most [Formula: see text] vertices or deleting at most [Formula: see text] edges. The construction shows that this is best possible.

Abstract. Let [Formula: see text] be integers and let [Formula: see text] be pairwise disjoint sets with [Formula: see text] for [Formula: see text]. Define [Formula: see text] as the collection of all subsets [Formula: see text] of [Formula: see text] with [Formula: see text] for each [Formula: see text]. In this paper, we show that if the matching number of [Formula: see text] is at most [Formula: see text] and [Formula: see text] for all [Formula: see text], then [Formula: see text]. Let [Formula: see text] with [Formula: see text] for all [Formula: see text]. We also prove that if [Formula: see text] do not contain a rainbow matching, then there exists [Formula: see text] in [Formula: see text] such that [Formula: see text].