Lower bounds for Ramsey numbers: $\mathrm{R}(6,8)\ge 135$ and $\mathrm{R}(8,10)\ge 345$

Source abstract

We prove the lower bounds $\mathrm{R}(6,8)\ge 135$ and $\mathrm{R}(8,10)\ge 345$, improving the bounds 134 and 343 listed in the April 2026 revision of Radziszowski's dynamic survey. We give explicit red/blue colorings of $K_{134}$ with no red $K_6$ or blue $K_8$, and of $K_{344}$ with no red $K_8$ or blue $K_{10}$, and verify them with two independently written exhaustive clique checkers. Starting from published colorings, we find these witnesses by adding vertices and repairing the resulting monochromatic cliques through local search. The search uses exact conflict counts, preparation moves aimed at making remaining cliques cheaper to break, and mutations followed by repair. A loss based on the largest monochromatic clique containing each edge yielded a useful intermediate state for the $\mathrm{R}(6,8)$ construction. Guided by the author, an AI coding agent wrote and ran the search code; we describe the methods and document the ancestry of the resulting colorings.