A family F is t-intersecting if any two members have at least t common elements. Erdos, Ko and Rado (1961) proved that the maximum size of a t-intersecting family of subsets of size k is equal to ((n-t)(k-t) ) if n >= n(0)(k, t). Alon, Aydinian and Huang (2014) considered families generalizing intersecting families, and proved the same bound. In this paper, we give a strengthening of their result by considering families generalizing t-intersecting families for all t >= 1. In 2004, Talbot generalized Bollobas's Two Families Theorem (Bollobas, 1965) to t-intersecting families. In this paper, we proved a slight generalization of Talbot's result by using the probabilistic method. (C) 2015 Elsevier Ltd. All rights reserved.
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