Unbalancing sets and an almost quadratic lower bound for syntactically multilinear arithmetic circuits

Abstract

We prove a lower bound of ω(n2/log2n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x1,...,xn). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([31]), who proved a lower bound of ω(n4/3/log2n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.