The $d$-chromatic Ramsey number for stars
Aijun Yi, Zhidan Luo
TL;DR
The paper extends Ramsey theory to the $d$-chromatic setting by defining $r^{s,t}(G_{1},\dots,G_{c})$ and its star-critical variant $r_{*}^{s,t}$, where color-sets $A_i$ of size $s$ come from a $t$-coloring. It develops a decomposition-based framework using $1$-factor, $2$-factor, and Hamiltonian-cycle decompositions to characterize thresholds for hosting copies of stars $K_{1,m_A}$ under color-set constraints, parameterized by $\ell_i$ and parity. The authors derive explicit formulas for $r^{s,t}(K_{1,m})$ and $r_{*}^{s,t}(K_{1,m})$, and obtain partial results for multi-star families $r^{s,t}(K_{1,n_1},\dots,K_{1,n_t})$, under natural consistency conditions on the $m_A$. These results generalize classical Ramsey theory to a refined, color-structure framework and connect to star-critical Ramsey theory in multi-color settings.
Abstract
In 1978, Chung and Liu generalized the definition of the Ramsey number. They introduced the $d$-chromatic Ramsey number as follows. Let $1\leq d< c$ be integers and let $A_{1}, \dots, A_{t}$ be subsets with size $d$ of $[c]$, where $t= {c\choose d}$. For given graphs $G_{1}, \dots, G_{t}$, {\it the $d$-chromatic Ramsey number}, $r^{d, c}(G_{1}, \dots, G_{t})$, is the minimum positive integer $N$ such that every $c$-coloring of $E(K_{N})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [t]$. The {\it star-critical $d$-chromatic Ramsey number}, $r_{*}^{d, c}(G_{1}, \dots, G_{t})$, is the minimum positive integer $k$ such that every $c$-coloring of $E(K_{N}- K_{1, N- 1- k})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [t]$, where $N= r^{d, c}(G_{1}, \dots, G_{t})$. If $G_{1}, \dots, G_{t}= G$, then we simplify them as $r^{d, c}(G)$ (it also call {\it the weakened Ramsey number}) and $r^{d, c}_{*}(G)$, respectively. In this paper, we determine all the value of $r^{d, c}(K_{1, n})$, $r_{*}^{d, c}(K_{1, n})$ and part of the value of $r^{d, c}(K_{1, n_{1}}, \dots, K_{1, n_{t}})$.