On a problem in elementary number theory and a combinatorial problem

Abstract

Recently, I observed that using a combinatorial theorem due to Rado and myself (1) can be considerably improved and it might, in fact, be possible to obtain the correct order of magnitude for ft(n). The combinatorial theorem in question states as follows [2]: Let g(k, t) be the smallest integer so that if A1 , ***, As, s = g (k, t), are sets each having k or fewer elements then there are always t A's Ail, , Ai, which have pairwise the same intersection. We have (2) g(k, t) < k!(t 1)k+1. We conjectured that (2) can be improved to (cl ,c2, are absolute constants) (3) g(k, t) < Clk(t 1)k+l. The conjectured (3) would have applications to several questions in number theory. It is not difficult to show that lim g(k, t)1Ik k=oo