Monochromatic triangle-tilings in dense graphs without large independent sets
Xinmin Hou, Xiangyang Wang, Zhi Yin
TL;DR
This work addresses the problem of forcing large weak monochromatic triangle-tilings in dense, 2-edge-coloured graphs with sublinear independence number. It combines a Ramsey–Tilings viewpoint with the degree-form Regularity Lemma, constructing a reduced graph to guide a skeleton of $F_2$-structures and then patching local tilings with a collection of new lemmas that guarantee large monochromatic $K_3$-tilings inside five-block configurations. The main result delivers asymptotically optimal bounds: a tiling of size at least $|\Gamma| \ge 2\delta(G)-n-o(n)$ for $\frac12 n \le \delta(G) \le \frac35 n$, and at least $|\Gamma| \ge \delta(G)/3-o(n)$ for $\delta(G) > \frac35 n$. The paper extends prior work on triangle-factors and Ramsey–Turán problems under independence-number constraints, providing a robust technique for assembling large monochromatic structures in coloured dense graphs.
Abstract
Given two graphs $H$ and $G$, an $H$-tiling is a family of vertex-disjoint copies of $H$ in $G$. A perfect $H$-tiling covers all vertices of $G$. The Corradi-Hajnal theorem (1963) states that an $n$-vertex graph $G$ with minimum degree $δ(G)\ge 2n/3$ contains a perfect triangle-tiling. For an $n$-vertex graph $G$ with independence number $α(G)=o(n)$, Balogh, Molla and Sharifzadeh (Random Structures & Algorithms, 2016) showed that a minimum degree of $(\frac12+o(1))n$ forces a perfect triangle-tiling. In a 2-edge-colored graph, Balogh, Freschi, Treglown (European J. Combin. 2026) determined the (asymptotic) minimum degree threshold for forcing a strong or weak monochromatic triangle-tiling covering a prescribed proportion of the vertices: a strong tiling requires all triangles to be in the same color class, while a weak tiling only requires each triangle to be monochromatic. In this paper, we combine the conditions from these two lines of work and prove that every $2$-edge-colored $n$-vertex graph $G$ with $α(G)=o(n)$ contains a weak monochromatic triangle-tiling $Γ$ of size \[ |Γ|\ge \begin{cases} 2δ(G)-n-o(n), & \text{if }\frac12 n\le δ(G)\le \frac35 n,\\[2mm] δ(G)/3-o(n), & \text{if }δ(G)>\frac35 n. \end{cases} \] Both bounds are asymptotically optimal. We use the degree form regularity lemma in our proof.