A Combinatorial Theorem on Systems of Sets

It is quite easy to see that (2) and (3) give best possible results. Upper estimates for n were also given in [2] if either (Alt..., An)eS(k, </, m) and /<^(m+l) or if (Alt..., An)eS(k, I, m) and l^^(m+k), but the results in these cases are not best possible. Other results of this kind have been established in [3] and [4] which settle certain conjectures made in [2]. In this note we establish the following generalisation of Sperner's result (2). THEOREM 1. / / (A t , . . . , A„) e S(k, < w, m) then