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Polynomials that Sign Represent Parity and Descartes' Rule of Signs
Computational Complexity
Learning read-constant polynomials of constant degree modulo composites
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
An introduction to randomness extractors
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
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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 Erd 陆os' probabilistic argument shows the existence of graphs on vertices with no clique or independent set of size , the best explicit constructions achieve a far weaker bound. Constructing Ramsey graphs is closely related to polynomial representations of Boolean functions; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17]. 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 [17] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(\sqrt 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.