Explicit construction of RIP matrices is Ramsey-hard
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Matrices Φ∈^n× p satisfying the Restricted Isometry Property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for n=^O(1)p, the explicit construction of such matrices defied the repeated efforts, and the most known approaches hit the so-called √(n) sparsity bottleneck. The notable exception is the work by Bourgain et al bourgain2011explicit constructing an n× p RIP matrix with sparsity s=Θ(n^1 2+ϵ), but in the regime n=Ω(p^1-δ). In this short note we resolve this open question in a sense by showing that an explicit construction of a matrix satisfying the RIP in the regime n=O(^2 p) and s=Θ(n^1 2) implies an explicit construction of a three-colored Ramsey graph on p nodes with clique sizes bounded by O(^2 p) -- a question in the extremal combinatorics which has been open for decades.
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