Beyond the Erdös-Ko-Rado theorem

Abstract The exact bound in the Erdős-Ko-Rado theorem is known [F, W]. It states that if n ⩾ (t + 1)(k − t + 1), and F is a t-intersecting family of k-sets of an n-set (|F ∩ F′| ⩾ t for all F, F′ ∈ F ),then | F ⩽( n−1 k−1 ) . Define A r = {F ⊂ {1, 2, …, n} : |F| = k, |F ∩ {1, 2, …, t + 2r}| ⩾ t + r}. Here it is proved that for n>c t log (t+1) (k − t + 1) one has | F | ⩽ maxr | A r|.