On set systems with restricted intersections modulo a composite number

Extended Abstract
  • Vince Grolmusz
Session 3: Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1276)


Let S be a set of n elements, and let H be a set-system on S, which satisfies that the size of any element of H is divisible by m but the intersection of any two elements of H is not divisible by m If m is a prime or prime-power, then the famous Frankl-Wilson theorem [3] implies that |H| = O(nm-1), i.e. for fixed m, its size is at most polynomial in n. This theorem has numerous applications in combinatorics and also in geometry, (c.f. the disproof of Borsuk's conjecture by Kahn and Kalai in 1992 ([4]), or explicit constructions of Ramsey graphs, or other geometric applications related to the Hadwiger-problem.) Frankl and Wilson asked in [3] whether an analogous upper bound existed for non-prime power, composite moduli. Here we show a surprising construction of a superpolynomial-sized uniform set-system H satisfying the intersection-property, for every non-prime-power, composite m, negatively settling a related conjecture of Babai and Frankl [1]. The proof uses a low-degree polynomial-construction of Barrington, Beigel and Rudich [2], and a new method (Lemma 8), for constructing set-systems from multivariate polynomials.


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  1. 1.
    L. Babai and P. Frankl, Linear algebra methods in combinatorics, Department of Computer Science, The University of Chicago, September 1992. preliminary version.Google Scholar
  2. 2.
    D. A. M. Barrington, R. Beigel, and S. Rudich: Representing Boolean functions as polynomials modulo composite numbers. Comput. Complexity, 4 (1994), pp. 367–382. Appeared also in Proc. 24th Ann. ACM Symp. Theor. Comput., 1992.Google Scholar
  3. 3.
    P. Frankl and R. M. Wilson: Intersection theorems with geometric consequences. Combinatorica, 1 (1981), pp. 357–368.Google Scholar
  4. 4.
    J. Kahn and G. Kalai: A counterexample to Borsuk's conjecture. manuscript, 1992.Google Scholar
  5. 5.
    D. K. Ray-Chaudhuri and R. M. Wilson: On t-designs. Osaka J. Math., 12 (1975), pp. 735–744.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Vince Grolmusz
    • 1
  1. 1.Department of Computer ScienceEötvös UniversityBudapestHungary

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