Weighted arcs,the finite radon transform and a Ramsey problem

We establish a link between the theory of (k, v)-arcs in affine planes and a graph theoretic Ramsey problem: A (n, k)-coloring of the complete graphK_u is a coloring of the edges ofK_u withk colours such that monochromatic connected subgraphs have at mostn vertices. The Ramsey numberf(n, k) is the smallestu such thatK_u does not admit a (n, k)-coloring. Let be an affine plane of orderq. Aweighted v-arc is a function w: such that line integrals are at mostv. Themass ofw is the sum of all the weights.In certain cases the maximum massM_q(v) can be determined (maximum over all weightedv-arcs and all affine planes of orderq). This yields the exact value off(n, q + 1) for all large enoughn.