Abstract: In this talk we shall discuss a connection between some conjectures in anticoncentration theory for sums of random vectors by Lee Jones (1978), Leader-Radcliffe (1994) and the Strong Perfect Graph Theorem. We will use the latter result to obtain a useful decomposition of distributions in question to establish both conjectures up to a 1+o(1) multiplicative factor for finite dimensional vector spaces. The talk is based on recent joint work with V. Kurauskas (Vilnius University).