Given graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

中文翻译：

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Komlós 在随机图中的平铺定理

给定图表G和H，一组顶点不相交的副本H在G被称为H平铺. Conlon、Gowers、Samotij 和 Schacht 表明，对于给定的图H和一个常数γ>0， 那里存在C>0这样如果$p \ge C{n^{ - 1/{m_2}(H)}}$, 然后几乎可以肯定地渐近每个跨越子图G随机图的𝒢(n, p)至少具有最低学位$\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$包含一个H- 覆盖所有但最多的平铺γn顶点。这里，χcr(H)表示临界色数，由 Komlós 引入的参数，以及米2(H) 是 2-密度的H. 我们证明了这个定理可以被引导以获得一个H- 平铺覆盖所有但最多$\gamma {(C/p)^{{m_2}(H)}}$顶点，当$p \ge C{n^{ - 1/{m_2}(H)}}$. 在这种情况下H=ķ3，这回答了 Balogh、Lee 和 Samotij 的问题。此外，对于任意图H我们给出一个上限p对于其中一些剩余是不可避免的，并且限制在最大的大小上H- 平铺p低于这个值。