Chromatic Ramsey numbers and two-color Turán densities
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| Format: | Preprint |
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2024
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| _version_ | 1866910601159114752 |
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| author | Axenovich, Maria Gaa, Simon Liu, Dingyuan |
| author_facet | Axenovich, Maria Gaa, Simon Liu, Dingyuan |
| contents | Given a graph $G$, its $2$-color Turán number $\mathrm{ex}^{(2)}(n,G)$ is the largest number of edges in an $n$-vertex graph whose edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $π^{(2)}(G)=\lim_{n\to\infty}\mathrm{ex}^{(2)}(n,G)/\binom{n}{2}$ be the $2$-color Turán density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Turán density of some graph? It is known that $π^{(2)}(G)=1-(R_χ(G)-1)^{-1}$, where $R_χ(G)$ is the chromatic Ramsey number of $G$. However, determining specific values of $R_χ(G)$ is challenging.
Burr, Erdős, and Lovász showed that $(k-1)^2+1\leqslant{R_χ(G)}\leqslant{R(k)}$, for any $k$-chromatic graph $G$, where $R(k)$ is the classical Ramsey number. The upper bound here can be attained by a clique and the lower bound is achieved by a graph constructed by Zhu. To the best of our knowledge, there are no other, besides these two, known values of $R_χ(G)$ among $k$-chromatic graphs $G$ for general $k$. In this paper we prove that there are $Ω(k)$ different values of $R_χ(G)$ among $k$-chromatic graphs $G$. In addition, we determine a new value for the chromatic Ramsey numbers of $4$-chromatic graphs. This sheds more light into the possible $2$-color Turán densities of graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_07535 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chromatic Ramsey numbers and two-color Turán densities Axenovich, Maria Gaa, Simon Liu, Dingyuan Combinatorics Given a graph $G$, its $2$-color Turán number $\mathrm{ex}^{(2)}(n,G)$ is the largest number of edges in an $n$-vertex graph whose edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $π^{(2)}(G)=\lim_{n\to\infty}\mathrm{ex}^{(2)}(n,G)/\binom{n}{2}$ be the $2$-color Turán density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Turán density of some graph? It is known that $π^{(2)}(G)=1-(R_χ(G)-1)^{-1}$, where $R_χ(G)$ is the chromatic Ramsey number of $G$. However, determining specific values of $R_χ(G)$ is challenging. Burr, Erdős, and Lovász showed that $(k-1)^2+1\leqslant{R_χ(G)}\leqslant{R(k)}$, for any $k$-chromatic graph $G$, where $R(k)$ is the classical Ramsey number. The upper bound here can be attained by a clique and the lower bound is achieved by a graph constructed by Zhu. To the best of our knowledge, there are no other, besides these two, known values of $R_χ(G)$ among $k$-chromatic graphs $G$ for general $k$. In this paper we prove that there are $Ω(k)$ different values of $R_χ(G)$ among $k$-chromatic graphs $G$. In addition, we determine a new value for the chromatic Ramsey numbers of $4$-chromatic graphs. This sheds more light into the possible $2$-color Turán densities of graphs. |
| title | Chromatic Ramsey numbers and two-color Turán densities |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.07535 |