Constructing Ramsey graphs from Boolean function representations

Abstract

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2n, the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18].

We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1}n except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials.

Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.

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References

  1. [1]

    N. Alon and R. Beigel: Lower bounds for approximations by low degree polynomials over ℤm, Proceedings of the 16th IEEE Conference on Computational Complexity (CCC), 2001.

    Google Scholar 

  2. [2]

    N. Alon: The Shannon capacity of a union, Combinatorica, 18 (1998), 301–310.

    Article  MATH  MathSciNet  Google Scholar 

  3. [3]

    B. Barak: A simple explicit construction of an n õ(log n) Ramsey graph, arXiv.org, math.CO/0601651, 2006.

  4. [4]

    D. A. Barrington, R. Beigel and S. Rudich: Representing Boolean functions as polynomials modulo composite numbers, Computational Complexity, 4 (1994), 367–382.

    Article  MATH  MathSciNet  Google Scholar 

  5. [5]

    L. Babai and P. Frankl: Linear Algebra Methods in Combinatorics, Preliminary version 2, 1992.

    Google Scholar 

  6. [6]

    L. Babai, P. Frankl, S. Kutin, and D. Štefankovič Set systems with restricted intersections modulo prime powers, Journal of Combinatorial Theory, Ser. A, 95 (2001).

  7. [7]

    N. Bhatnagar, P. Gopalan and R. J. Lipton: Symmetric polynomials over ℤm and simultaneous communication protocols, Proceedings of the 44 th Annual Symposium on the Foundations of Computer Science (FOCS), 2003.

    Google Scholar 

  8. [8]

    B. Barak, A. Rao, R. Shaltiel and A. Wigderson: 2-source dispersers for n o(1) entropy and Ramsey graphs beating the Frankl-Wilson construction, in: Proceedings of the 38th Symposium on Theory of Computing (STOC), 2006.

    Google Scholar 

  9. [9]

    Z. Dvir, P. Gopalan and S. Yekhanin: Matching vector codes, in: Proceedings of the 51 st Annual Symposium on the Foundations of Computer Science (FOCS), 2010.

    Google Scholar 

  10. [10]

    K. Efremenko: 3-query locally decodable codes of subexponential length, in: 41st ACM Symposium on Theory of Computing (STOC), 39–44, 2009.

    Google Scholar 

  11. [11]

    P. Erdös: Some remarks on the theory of graphs. Bulletin of the A. M. S., 53 (1947), 292–294.

    Article  MATH  Google Scholar 

  12. [12]

    P. Frankl and R. Wilson: Intersection theorems with geometric consequences, Combinatorica, 1 (1981), 357–368.

    Article  MATH  MathSciNet  Google Scholar 

  13. [13]

    P. Gopalan: Constructing Ramsey graphs from Boolean function representations, in: Proceedings of the 21st IEEE Conference on Computational Complexity (CCC), 2006.

    Google Scholar 

  14. [14]

    P. Gopalan: Query-efficient algorithms for polynomial interpolation over composites, in: Proceedings of the ACM-SIAM Symposium on Discrete algorithms (SODA), 2006.

    Google Scholar 

  15. [15]

    F. Green: Complex Fourier technique for lower bounds on the mod-m degree, Computational Complexity, 9 (2000), 16–38.

    Article  MATH  MathSciNet  Google Scholar 

  16. [16]

    V. Grolmusz: On the weak mod m representation of Boolean functions. Chicago Journal of Theoretical Computer Science, 2 (1995).

  17. [17]

    V. Grolmusz: On set systems with restricted intersections modulo a composite number, in: COCOON, pages 82–90, 1997.

    Google Scholar 

  18. [18]

    V. Grolmusz: Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs, Combinatorica, 20 (2000), 71–86.

    Article  MATH  MathSciNet  Google Scholar 

  19. [19]

    V. Grolmusz: Constructing set systems with prescribed intersection sizes, Journal of Algorithms, 44 (2002), 321–337.

    Article  MATH  MathSciNet  Google Scholar 

  20. [20]

    G. H. Hardy and E. M. Wright: An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1985.

    Google Scholar 

  21. [21]

    S. Kutin: Constructing large set systems with given intersection sizes modulo composite numbers, Combinatorics, Probability and Computing, 11 (2002).

  22. [22]

    M. Naor: Constructing Ramsey graphs from small probability spaces, Manuscript, available online, 1993.

    Google Scholar 

  23. [23]

    P. Raghavendra: A note on Yekhanin’s locally decodable codes, in: Electronic Colloquium on Computational Complexity (ECCC), TR07-016, 2007.

    Google Scholar 

  24. [24]

    R. Smolensky: On representations by low-degree polynomials, in: Proceedings of the 34 th Annual Symposium on Foundations of Computer Science (FOCS), 1993.

    Google Scholar 

  25. [25]

    G. Tardos and D. Barrington: A lower bound on the mod 6 degree of the OR function, Computational Complexity, 7 (1998), 99–108.

    Article  MATH  MathSciNet  Google Scholar 

  26. [26]

    S.-C. Tsai: Lower bounds on representing Boolean functions as polynomials in ℤm, SIAM Journal of Discrete Mathematics, 9 (1996), 55–62.

    Article  MATH  Google Scholar 

  27. [27]

    R. Williams: Non-uniform acc circuit lower bounds. in: 26th IEEE Conference on Computational Complexity (CCC 2011), 2011.

    Google Scholar 

  28. [28]

    S. Yekhanin: Towards 3-query locally decodable codes of subexponential length, Journal of the ACM, 55 (2008), 1–16.

    Article  MathSciNet  Google Scholar 

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Correspondence to Parikshit Gopalan.

Additional information

A preliminary version of this paper appeared in CCC’06 [13].

Work done while the author was at Georgia Tech and the University of Texas at Austin.

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Gopalan, P. Constructing Ramsey graphs from Boolean function representations. Combinatorica 34, 173–206 (2014). https://doi.org/10.1007/s00493-014-2367-1

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Mathematics Subject Classification (2000)

  • 05C55
  • 05C50