English–McCourt–Mattes–Phillips cocktail-party graph conjecture

Let $G^c=(V,E)$ be a cocktail party graph, obtained from a complete graph on an even number of vertices by deleting a perfect matching. For a color $i\in[2]$, let $G^c_i$ be the spanning subgraph consisting of the edges of color $i$, and call a subset a diameter $2$ subset of $G^c_i$ when every two of its vertices have distance at most two in the induced subgraph.

English–McCourt–Mattes–Phillips conjecture. In every $2$-coloring of the edges of $G^c$, there exist $A,B\subseteq V$ and colors $i,j\in[2]$ such that

$$A\cup B=V$$

and $A$ and $B$ are diameter $2$ subsets of $G^c_i$ and $G^c_j$, respectively.

This conjecture asks for a two-set strengthening of the component-cover phenomenon in the first conjecture. The paper presents it as unresolved and studies related 2-reachable coverings.