Improved explicit upper bounds on the Ramsey numbers guaranteeing monochromatic solutions to x1+...+x_{m+1}=y1+...+ym in r-colorings of [N], with the bound N>(2m+1)^r (r!)^{1/m} and optimality of N=2^r for some m.
Koścuiszko,Schur-like numbers and a lemma of Shearer(2025), preprint available at https://arxiv.org/abs/2507.216562, 5, 10
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Improved Ramsey bounds for generalized Schur equations
Improved explicit upper bounds on the Ramsey numbers guaranteeing monochromatic solutions to x1+...+x_{m+1}=y1+...+ym in r-colorings of [N], with the bound N>(2m+1)^r (r!)^{1/m} and optimality of N=2^r for some m.