Year: 2026
Type: preprint
Abstract: For ordered graphs $H_1,\ldots,H_t$, let $\rt(H_1,\ldots,H_t)$ denote the least integer $N$ such that every $t$-coloring of the edges of the naturally ordered complete graph on $[N]$ contains an ordered copy of $H_i$ in color $i$ for some $i\in[t]$. We prove that a uniformly random ordered matching $M$ on $n$ vertices with interval chromatic number two asymptotically almost surely satisfies \[ \rt(K_3,M) =Ω\left(\frac{n^{4/3}}{(\log n)^{1/3}}\right). \] This strengthens the lower bound $Ω((n/\log n)^{5/4})$ of Balko and Poljak for such random matchings and improves the general existential lower bound of Conlon, Fox, Lee and Sudakov by a factor of $\log n$. The proof combines pseudorandom triangle-free graphs, a coarse encoding of order-preserving embeddings, and a permutation avoidance estimate derived from Brègman's inequality.
Source: arXiv (Cornell University)
Authors: Chen, Wen, Lin, Qizhong, You, Chunlin
Institutions: Fuzhou University, Yancheng Teachers University
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213602569
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