Paper detail

On Komlós' tiling theorem in random graphs

Conlon, Gowers, Samotij, and Schacht showed that for a given graph $H$ and a constant $γ> 0$, there exists $C > 0$ such that if $p \ge Cn^{-1/m_2(H)}$ then asymptotically almost surely every spanning subgraph $G$ of the random graph $\mathcal{G}(n,p)$ with minimum degree at least $δ(G) \ge (1 - 1/χ_{\mathrm{cr}}(H) + γ)np$ contains an $H$-packing that covers all but at most $γn$ vertices. Here, $χ_{\mathrm{cr}}(H)$ denotes the critical chromatic threshold, a parameter introduced by Komlós. We show that this theorem can be bootstraped to obtain an $H$-packing covering all but at most $γ(C/p)^{m_2(H)}$ vertices, which is strictly smaller when $p > C n^{-1/m_2(H)}$. In the case where $H = K_3$ this answers the question of Balogh, Lee, and Samotij. Furthermore, we give an upper bound on the size of an $H$-packing for certain ranges of $p$.