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 (), 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. 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  and Enomoto et al. , who proved in 1987 the above statement (with a tight constant) for odd values of n, leaving the even case open.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics