On a Variation of Schur Numbers

For every mV 3, let n ˆ R…L3…m†† be the least integer such that for every 2-coloring of the set S ˆ f1; 2; . . . ; ng, there exists in S a monochromatic solution to the following system. L3…m†: x1 ‡ x2 ‡ ‡ xmÿ1 < xm x1 < x2 < < xm: The main result of this paper is that R…L3…m†† ˆ 9 16 m ÿm2 ‡m‡ 1 if m1 0 …mod 4† 9 16 m ÿm2 ‡ 13 16 m‡ 13 8 if m1 1 …mod 4† 9 16 m ÿm2 ‡m‡ 1 2 if m1 2 …mod 4† 9 16 m ÿm2 ‡ 17 16 m‡ 5 8 if m1 3 …mod 4† 8>>><>>>>>>: Moreover, it is shown that, up to a switching of the colors, there exists a unique 2-coloring of the set f1; 2; . . . ;R…L3…m†† ÿ 1g that avoids a monochromatic solution to the above system.