4th Irish 1991 Problem A4 8 people decide to hold daily meetings subject to the following rules. At least one person must attend each day. A different set of people must attend on different days. On day N for each 1 ≤ k < N, at least one person must attend who was present on day k. How many days can the meetings be held?
128 Solution We want the largest possible number of subsets of {1, 2, ... , 8} such that no two subsets are disjoint. The number cannot exceed 128 because for each subset X we cannot include both X and its complement. We can achieve 128 by taking all the subsets containing 1. 4th Irish 1991 © John Scholes jscholes@kalva.demon.co.uk 6 February 2004 Last updated/corrected 6 Feb 04