On the size edge-ordered Ramsey numbers of graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909975925751808 |
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| author | Song, Yanyan Mao, Yaping |
| author_facet | Song, Yanyan Mao, Yaping |
| contents | For edge-ordered graphs $G^{\prec}$ and $H^{\prec}$, the size edge-ordered Ramsey number $\hat{r}_{\text{edge}}(G^{\prec}, H^{\prec})$ is defined as the smallest integer $m$ for which there exists an edge-ordered graph $F^{\prec}$ (with underlying graph $F$) having $m$ edges, such that every $2$-coloring of the edges of $F^{\prec}$ contains a monochromatic edge-ordered subgraph isomorphic to $G^{\prec}$ or a monochromatic edge-ordered subgraph isomorphic to $H^{\prec}$. Fox and Li posed a foundational question: which families of edge-ordered graphs have linear or near-linear size edge-ordered Ramsey numbers? In this paper, we apply Szemerédi's regularity lemma to prove that, even for sparse graph families, specifically the well-defined class of edge-ordered book graphs, the size edge-ordered Ramsey numbers of this family exhibit non-linear growth. Furthermore, we show that three families of edge-ordered graphs exhibit linear or near-linear size edge-ordered Ramsey numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21647 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the size edge-ordered Ramsey numbers of graphs Song, Yanyan Mao, Yaping Combinatorics For edge-ordered graphs $G^{\prec}$ and $H^{\prec}$, the size edge-ordered Ramsey number $\hat{r}_{\text{edge}}(G^{\prec}, H^{\prec})$ is defined as the smallest integer $m$ for which there exists an edge-ordered graph $F^{\prec}$ (with underlying graph $F$) having $m$ edges, such that every $2$-coloring of the edges of $F^{\prec}$ contains a monochromatic edge-ordered subgraph isomorphic to $G^{\prec}$ or a monochromatic edge-ordered subgraph isomorphic to $H^{\prec}$. Fox and Li posed a foundational question: which families of edge-ordered graphs have linear or near-linear size edge-ordered Ramsey numbers? In this paper, we apply Szemerédi's regularity lemma to prove that, even for sparse graph families, specifically the well-defined class of edge-ordered book graphs, the size edge-ordered Ramsey numbers of this family exhibit non-linear growth. Furthermore, we show that three families of edge-ordered graphs exhibit linear or near-linear size edge-ordered Ramsey numbers. |
| title | On the size edge-ordered Ramsey numbers of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.21647 |