Independet Transversals in Sparse Partite Hypergraphs

An [n, k, r]-hypergraph is a hypergraph :Yf = (V, E) whose vertex set V is partitioned into n k-element sets V1, V2, ... , Vn and for which, for each choice of r indices, 1 :::;; i1 < i2 < ... < ir :::;; n, there is exactly one edge e E E such that len Vii = 1 if i E {i1, i2, ... , ir} and otherwise le n Vii = 0. An independent transversal of an [n, k, r ]-hypergraph is a set T = {a1,a2, .. . ,an}~ V such that ai E Vi fori= 1, 2, ... ,nand e 1:. T for all e E E. The purpose of this note is to estimate fr(k), defined as the largest n for which any [n,k,r]hypergraph has an independent transversal. The sharpest results are for r = 2 and for the case when k is small compared to r.