Forbidden Intersection Patterns in the Families of Subsets

holds, and this estimate is sharp as the family of all n2 -element subsets shows. There is a very large number of generalizations and analogues of this theorem. Here we will consider only results when the condition on F excludes certain configurations what can be expressed by inclusion, only. That is, no intersections, unions, etc. are involved. The first such generalization was obtained by Erdős [4]. The family of k distinct sets with mutual inclusions, F1 ⊂ F2 ⊂ . . . Fk is called a chain of lenght k. It will be simply denoted by Pk. Let La(n, Pk) denote the largest family F without a chain of lenght k. Theorem 1.2 ([4]). La(n, Pk+1) is equal to the sum of the k largest bimomial coefficients of order n. Let Vr denote the r-fork, that is the following family of distinct sets: F ⊂ G1, F ⊂ G2, . . . F ⊂ Gr. The quantity La(n, Vr), that is, the largest family on n elements containing no Vr was first (asymptotically) determined for r = 2. Theorem 1.3 ([7]). ( n n2 )( 1 + 1 n +O ( 1 n2 )) ≤ La(n, V2) ≤ ( n n2 )( 1 + 2 n ) .