On the largest size of a partition that is both s-core and t-core

TextLet s,ts,t be relatively prime positive integers. We prove a conjecture of Aukerman, Kane and Sze regarding the largest size of a partition that is simultaneously s-core and t-core by solving an equivalent problem concerning sets S   of positive integers with the property that for n∈Sn∈S, n−s∈Sn−s∈S whenever n⩾sn⩾s and n−t∈Sn−t∈S whenever n⩾tn⩾t.VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=o1OEug8LryU.