Geodesic
A note about monochromatic components in graphs of large minimum degree
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 607-618

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For all positive integers r≥ 3 and n such that r^2-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-2r^2-r)n-2 such that the largest monochromatic component has order less than nr-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.
Keywords: Ramsey theory, fractional matchings, block designs 
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DeBiasio, Louis; Krueger, Robert A. A note about monochromatic components in graphs of large minimum degree. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 607-618. http://geodesic.mathdoc.fr/item/DMGT_2023_43_3_a1/