Explicit construction of RIP matrices is Ramsey-hard

Abstract

Matrices $\Phi\in\R^{n\times p}$ 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 $n=\log^{O(1)}p$, the explicit construction of such matrices defied the repeated efforts, and the most known approaches hit the so-called $\sqrt{n}$ sparsity bottleneck. The notable exception is the work by Bourgain et al \cite{bourgain2011explicit} constructing an $n\times p$ RIP matrix with sparsity $s=\Theta(n^{{1\over 2}+\epsilon})$, but in the regime $n=\Omega(p^{1-\delta})$. 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 $n=O(\log^2 p)$ and $s=\Theta(n^{1\over 2})$ implies an explicit construction of a three-colored Ramsey graph on $p$ nodes with clique sizes bounded by $O(\log^2 p)$ -- a question in the extremal combinatorics which has been open for decades.