For $1\le s<t$ and any graph $G$, the weakened Gallai-Ramsey number $gr^t_s(G)$ is defined to be the least $p\in \mathbb{N}$ such that every Gallai $t$-coloring of the edges of $K_p$ (i.e., a $t$-coloring that lacks rainbow triangles) contains a subgraph isomorphic to $G$ whose edges use at most $s$ of the colors. In the case of a book graph $B_n:=K_2+nK_1$, Jakhar and Moun determined the values $gr^3_2(B_3)=6$ and $gr^3_2(B_4)=7$. In this paper, we extend their results to $t>3$ colors, and we determine the values of $gr^3_2(B_n)$ for $5\le n\le 15$. General lower bounds for $gr^t_2(B_n)$ are also given.
For graphs $F$, $G$, and $H$, we write $F \to (G, H)$ if every red/blue edge-coloring of $F$ contains either a red copy of $G$ or a blue copy of $H$. The Ramsey number $R(G, H)$ is the smallest integer $N$ such that $K_N \to (G, H)$. Let $H := K_k + nK_h$ be the generalized book graph, and let $G := K_{p+1}(a_1, a_2, \...
The \emph{ordered Ramsey number} $r_<(G,H)$ of ordered $k$-graphs $G$ and $H$ is the least integer $N$ such that every red-blue edge-coloring of the naturally ordered complete $k$-graph on $[N]$ contains a blue ordered copy of $G$ or a red ordered copy of $H$. We prove that there is an absolute constant $c>0$ such that...
Wen Chen, Zi-Han He, Qi-Zhong Lin et al.· 0 citations
For a graph $G$ with $3\mid e(G)$, let $R(G,\mathbb Z_3)$ denote the least integer $N$ such that every edge weighting $w:E(K_N)\to\mathbb Z_3$ admits a copy of $G$ whose edge weights sum to zero. We prove that every $n$-vertex graph $G$ with $n\ge6$ and $3\mid e(G)$ satisfies $R(G,\mathbb Z_3)\le n+5$. Moreover, the st...
For $k, r, n$ natural numbers let $\rho_{r,k}(K_n)$ be the number of $r$-edge-colorings of $K_n$ that do not contain a rainbow copy of a $K_k$, that is, a copy of $K_k$ in which all edges receive different colors. When $k=3$, the quantity $\rho_{r,3}(K_n)$ represents the number of Gallai Colorings. It was proved by Bal...
For positive integers $t$ and $q$ let $b_q(t)$ be the smallest integer $n$ so that any coloring of the edges of the complete bipartite graph $K_{n,n}$ with $q$ colors yields a monochromatic copy of $K_{t,t}$. We give an independent proof that $b_2(t)\le 128\cdot 2^t$ for every positive integer $t$. More generally, for...
For a graph $H$, the anti-Ramsey number $\operatorname{ar}(n,H)$ is the maximum number of colors in an edge-coloring of $K_n$ containing no rainbow copy of $H$, where a copy is rainbow if its edges have pairwise distinct colors. Let $s,t$ be nonnegative integers with $s+t\ge2$, and let $H_{s,t}$ be a graph consisting o...
Hao-Jie Zheng· 0 citations
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