Sharp bounds for covering with large cliques and independent sets

Let $n(k_1,k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that $n(k,k)\ge 4k-O(k^{\frac{2}{3}})$, and conjectured that $n(k,k)=4k-4$. We prove this conjecture, and also establish the optimal lower bound in the more general case where $k_1$ and $k_2$ are arbitrary. We further consider the generalisation of the problem to $r$-edge-coloured complete graphs in which every vertex is contained in a size-$k$ monochromatic clique of each colour, and obtain upper and lower bounds on the size of such graphs.