@article{885ed4a8d4e34a2c8aedfbb65521330d,

title = "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 ([34]), 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 R{\"o}dl [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.",

keywords = "03D15, 68Q17, 68R05, 68W30",

author = "Noga Alon and Mrinal Kumar and Volk, {Ben Lee}",

note = "Publisher Copyright: {\textcopyright} 2020, J{\'a}nos Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.",

year = "2020",

month = apr,

day = "1",

doi = "10.1007/s00493-019-4009-0",

language = "אנגלית",

volume = "40",

pages = "149--178",

journal = "Combinatorica",

issn = "0209-9683",

publisher = "Janos Bolyai Mathematical Society",

number = "2",

}