On a conjecture of Alon

TextLet f(n,m)f(n,m) be the cardinality of largest subset of {1,2,…,n}{1,2,…,n} which does not contain a subset whose elements sum to m. In this note, we show thatf(n,m)=(1+o(1))nsnd(m) for all n(logn)1+ϵ⩽m⩽n29log2n, where snd(m)snd(m) is the smallest integer that does not divide m. This proves a conjecture of Alon posed in [N. Alon, Subset sums, J. Number Theory 27 (2) (1987) 196–205].VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=UG-NhNyitoQ.