We present a polynomial-time algorithm that colors any 3-colorable $n$-vertex graph using $O(n^{0.19539})$ colors, improving upon the previous best bound of $\widetilde{O}(n^{0.19747})$ by Kawarabayashi, Thorup, and Yoneda [STOC 2024]. Our result constitutes the first progress in nearly two decades on SDP-based approaches to this problem. The earlier SDP-based algorithms of Arora, Chlamtáč, and Charikar [STOC 2006] and Chlamtáč [FOCS 2007] rely on extracting a large independent set from a suitably "random-looking" second-level neighborhood, under the assumption that the KMS algorithm [Karger, Motwani, and Sudan, JACM 1998] fails to find one globally. We extend their analysis to third-level neighborhoods. We then come up with a new vector $5/2$-coloring, which allows us to extract a large independent set from some third-level neighborhood. The new vector coloring construction may be of independent interest.

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