On Generalized Schur Numbers

Let L(t) represent the equation x1 + x2 + · · · + xt−1 = xt. For k > 1, 0 6 i 6 k − 1, and ti > 3, the generalized Schur number S(k; t0, t1, . . . , tk−1) is the least positive integer m such that for every k-colouring of {1, 2, . . . ,m}, there exists an i ∈ {0, 1, . . . , k − 1} such that there exists a solution to L(ti) that is monochromatic in colour i. In this paper, we report twenty-six previously unknown values of S(k; t0, t1, . . . , tk−1) and conjecture that for 4 6 t0 6 t1 6 t2, S(3; t0, t1, t2) = t2t1t0 − t2t1 − t2 − 1.