## Abstract

We prove a lower bound of *Ω*(*n*^{2}/log^{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 ([34]), who proved a lower bound of *Ω*(*n*^{4/3}/log^{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.

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 2*n* of a set *X* of size 4*n*, so that any subset of *X* of size 2*n* 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.

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## Acknowledgments

Part of this work was done while the second author was visiting Tel Aviv University. We thank Amir Shpilka for the visit, for many insightful discussions, and for comments on an earlier version of this text. We also thank Srikanth Srivinasan for allowing us to include his proof of Lemma 2.1 in this paper, and Andy Drucker and the anonymous reviewers for their suggestions and comments on earlier versions of the manuscript.

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## Additional information

Part of this work was done at the Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, USA.

Research supported in part by NSF grant DMS-1855464, ISF grant 281/17, BSF grant 2018267 and the Simons Foundation.

Part of this work was done at the School of Computer Science, Tel Aviv University, Tel Aviv, Israel. The research leading to these results has received funding from the Israel Science Foundation (grant number 552/16).

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Alon, N., Kumar, M. & Volk, B.L. Unbalancing Sets and An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits.
*Combinatorica* **40, **149–178 (2020). https://doi.org/10.1007/s00493-019-4009-0

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### Mathematics Subject Classfication (2010)

- 03D15
- 68R05
- 68Q17
- 68W30