Pigeonhole step for triangle Ramsey numbers: $R_k(3) \le N$ implies $R_{k+1}(3) \le (k+1)(N-1) + 2$

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Pigeonhole step for triangle Ramsey numbers: $R_k(3) \le N$ implies $R_{k+1}(3) \le (k+1)(N-1) + 2$ · Prove2Me