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A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have atleast $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum sizeof a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t}\choose {k-t}}$ if $n\geq n_0(k,t)$. Alon, Aydinian, and Huang consideredfamilies generalizing intersecting families, and proved the same bound. In thispaper, we give a strengthening of their result by considering familiesgeneralizing $t$-intersecting families for all $t \geq 1$. In 2004, Talbotgeneralized Bollob\'{a}s's Two Families Theorem to $t$-intersecting families.In this paper, we proved a slight generalization of Talbot's result by usingthe probabilistic method.

On the Erdos-Ko-Rado Theorem and the Bollobas Theorem for t-intersecting families