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.
The ramsey number N(3, 3, 3, 3; 2)
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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.