A note about monochromatic components in graphs of large minimum degree

For all positive integers r ≥ 3 and n such that r2 − r divides n and an affine plane of order r exists, we construct an r-edge colored graph on n vertices with minimum degree (1−r-2r2-r{{r - 2} \over {{r^2} - r}})n−2 such that the largest monochromatic component has order less than nr-1{n \over {r - 1}}. This generalizes an example of Guggiari and Scott and, independently, Rahimi for r = 3 and thus disproves a conjecture of Gyárfás and Sárközy for all integers r ≥ 3 such that an affine plane of order r exists.