Largest Families Without an r -Fork

There is a very large number of generalizations and analogues of this theorem. Here we will mention only some 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 [3]. 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.