Unbalancing Sets and An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

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,...,x_n). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([34]), who proved a lower bound of Ω(n^(4/3)/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. A special case of our combinatorial result implies, for every n, a tight Ω(n) lower bound on the minimum size of a family F of subsets of cardinality 2n of a set X of size 4n, so that any subset of X of size 2n has intersection of size exactly n with some member of F. This settles a problem of Galvin up to a constant factor, extending results of Frankl and Rödl [15] and Enomoto et al. [12], who proved in 1987 the above statement (with a tight constant) for odd values of n, leaving the even case open