Defect Sauer results

In this paper we present a unified account of various results concerning traces of set systems, including the original lemma proved independent by Sauer [J. Combin. Theory Ser. A 13 (1972), 145–147], Shelah [Pacific J. Math 41 (1972), 271–276], and Vapnik and Chervonenkis [Theory Probab. Appl. 16 (1971), 264–280], and extend these results in various directions. Included are a new criterion for a set system to be extremal for the Sauer inequality and upper and lower bounds, obtained by random methods, for the trace of a set system of size nr guaranteed on some αn-sized subset of {1,2, …, n}.