Minimum Number of Monochromatic Subgraphs of a Random Graph

We consider the problem of minimizing the number of monochromatic subgraphs of a random graph, when each node of the host graph is assigned one of the two colors. Using a recently discovered contiguity between appearance of strictly balanced subgraphs $F$ in a random graph, and random hypergraphs where copies of $F$ are generated independently, we show that the minimum value converges to a limit, when the expected number of copies of $F$ is linear in the number of nodes $\mathrm{\mid }V\mathrm{\mid }$. Furthermore, using the connections with mean field spin glass models, we obtain an asymptotic expression for this limit as the normalized expected number of copies of $F$ and the size of $F$ diverge to infinity.