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Explicit two-source extractors and resilient functions. (English) Zbl 1419.05109
Given a positive integer $$k$$ and a positive real number $$\varepsilon$$, let $$K = 2^{k}$$. A (balanced) bipartite graph containing $$N$$ “left” vertices and $$N$$ “right” vertices is called a $$(k, \varepsilon)$$-two-source extractor if every subgraph with $$K$$ left vertices and $$K$$ right vertices contains $$(1/2 \pm \varepsilon)K^{2}$$ edges.
The main result of this work is the following.
Theorem 1. There is a positive constant $$C$$ such that for any natural number $$n$$, there is an explicit construction of a $$(k, \varepsilon)$$-two-source extractor on two sets of $$2^{n}$$ vertices with $$k = \log^{C}(n/\varepsilon)$$. The construction is explicit in the sense that there is an algorithm which runs in polynomial time $$\mathrm{poly}(n/\varepsilon)$$ that determines whether there is an edge between two nodes.
This result is applied in the proof of the following result.
Theorem 2. There is a positive constant $$C$$ such that for any natural number $$n$$, there is an explicit construction of bipartite $$K$$-Ramsey graphs on $$2N$$ vertices and a Ramsey graph on $$N$$ vertices where $$N = 2^{n}$$ and $$K = 2^{(\log \log N)^{C}}$$.
The proof utilizes a variety of tools from the theory of extractors and probability along with two technical “key lemmas”. The proof of the main result is completed by first using non-malleable extractors to reduce the construction of a two-source extractor to the problem of constructing resilient functions. Such a function is constructed to be computable by a polynomial sized constant depth monotone circuit.

##### MSC:
 05C35 Extremal problems in graph theory 68R05 Combinatorics in computer science
##### Keywords:
Ramsey graphs; randomness extractors
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##### References:
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