Sharp bounds for covering with large cliques and independent sets

TL;DR

This paper proves a conjecture about the minimum size of graphs containing large cliques and independent sets, providing sharp bounds and extending to multi-colored complete graphs.

Contribution

It establishes the exact value of $n(k,k)$ and generalizes bounds to graphs with arbitrary clique and independent set sizes, including multi-colored cases.

Findings

Proved that $n(k,k)=4k-4$ confirming Feige and Pauzner's conjecture.

Derived optimal lower bounds for graphs with specified clique and independent set sizes.

Extended the analysis to $r$-edge-colored complete graphs with monochromatic cliques.

Abstract

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 ${n(k, k) \geq 4k-O(k^\frac{2}{3})}$, 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.