Erdős–Gyárfás–Pyber monochromatic component cover conjecture

Prove that, for every integer r≥2, every r-edge-colouring of the complete graph K_n permits the vertex set to be covered by at most r−1 monochromatic connected components.

Background

The paper introduces monochromatic covers and defines the covering parameter for r-edge-colourings of a graph as the least number of monochromatic components needed to cover all vertices in every such colouring. It then recalls a conjecture of Erdős, Gyárfás, and Pyber asserting the bound r−1 for complete graphs. This conjecture provides general Ramsey-theoretic context for the paper’s study of sharper covering bounds in dense balanced bipartite graphs.