Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems
Journal of Combinatorial Theory Series A - Series A
Representing Boolean functions as polynomials modulo composite numbers
Computational Complexity - Special issue on circuit complexity
A lower bound on the MOD 6 degree of the or function
Computational Complexity
Set systems with restricted intersections modulo prime powers
Journal of Combinatorial Theory Series A
Set-Systems with Restricted Multiple Intersections and Explicit Ramsey Hypergraphs
Set-Systems with Restricted Multiple Intersections and Explicit Ramsey Hypergraphs
Constructing Set-Systems with Prescribed Intersection Size
Constructing Set-Systems with Prescribed Intersection Size
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We consider k-uniform set systems over a universe of sizen such that the size of each pairwise intersection of setslies in one of s residue classes mod q, but kdoes not lie in any of these s classes. A celebrated theoremof Frankl and Wilson [8] states that any such set system has sizeat most (ns) when q isprime. In a remarkable recent paper, Grolmusz [9] constructed setsystems of superpolynomial size Ω(exp(clog2 n/log log n)) when q = 6. Wegive a new, simpler construction achieving a slightly improvedbound. Our construction combines a technique of Frankl [6] ofapplying polynomials to set systems with Grolmusz's idea ofemploying polynomials introduced by Barrington, Beigel and Rudich[5]. We also extend Frankl's original argument to arbitraryprime-power moduli: for any &egr; 0, we construct systemsof size ns+g(s), where g(s)= Ω(s1−Ε). Our work overlapswith a very recent technical report by Grolmusz [10].