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We prove a lower bound of Ω(n^2/^2 n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x_1, ..., x_n). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([RSY08]), who proved a lower bound of Ω(n^4/3/^2 n) 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.arXiv:1708.02037v3 fatcat:c6dnrpzmkvfitdytesarela4di