Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors

A distribution X over binary strings of length π has minentropy k if every string has probability at most 2 ^-k in X. ¹ We say that X is a δ-source if its rate k/n is at least δ, We give the following new explicit constructions (namely, poly(π)-time computable functions) of deterministic extractors, dispersers and related objects. All work for any fixed rate δ > 0, No previous explicit construction was known for either of these, for any δ < 1/2, The first two constitute major progress to very long-standing open problems 1. Bipartite Ramsey f ₁ : ({0, l} ⁿ) ² → {0, 1}, such that for any two independent δ-sources X ₁, X ₂ we have f ₁(X ₁,X ₂) = {0, 1}. This implies a new explicit construction of 2N-vertex bipartite graphs where no induced N ^δ by N ^δ subgraph is complete or empty. 2. Multiple source extraction f ₂ : ({0, l} ⁿ) ³ → {0, 1}, such that for any three independent 5-sources X ₁, X ₂, X ₃ we have that f ₂(X ₁, X ₂, X ₃) is (o(1)-close to being) an unbiased random bit, 3. Constant seed condenser ²f ₃ : {0, 1} ⁿ → ({0, l} ^m) ^c, such that for any δ-source X, one of the c output dis tributions f ₃(X) _i, is a 0.9-source over {0, l} ^m, Here c is a constant depending only on 6, 4. Subspace Ramsey f ₄ : {0, 1} ⁿ → {0, 1}, such that for any affine-δ-source ³X we have f ₄(X) = {0, 1}. The constructions are quite involved and use as building blocks other new and known gadgets, But we can point out two important themes which recur in these constructions, One is that gadgets which were designed to work with independent inputs, sometimes perform well enough with correlated, high entropy inputs, The second is using the input to (introspectively) find high entropy regions within itself.