A Note on the Multi-Colored Bipartite Ramsey Number for Paths

For bipartite graphs [Formula: see text], the bipartite Ramsey number [Formula: see text] is the least positive integer [Formula: see text] so that any coloring of the edges of [Formula: see text] with [Formula: see text] colors will result in a copy of [Formula: see text] in the [Formula: see text]th color for some [Formula: see text]. In this paper, we get the exact value of [Formula: see text], and obtain the upper and lower bounds of [Formula: see text], where [Formula: see text] denotes a path with [Formula: see text] vertices.

A Note on Restricted Online Ramsey Numbers of Matchings

Consider the following game between Builder and Painter. We take some families of graphs $\mathcal{G}_{1},\ldots,\mathcal{G}_t$ and an integer $n$ such that $n \geq R(\mathcal{G}_1,\ldots,\mathcal{G}_t)$. In each turn, Builder picks an edge of initially uncoloured $K_n$ and Painter colours that edge with some colour $i \in \left\{ 1,\ldots,t \right\}$ of her choice. The game ends when a graph $G_i$ in colour $i $ for some $G_i \in \mathcal{G}_i$ and some $i$ is created. The restricted online Ramsey number $\tilde{R}(\mathcal{G}_{1},\ldots,\mathcal{G}_t;n)$ is the minimum number of turns that Builder needs to guarantee the game to end. In a recent paper, Briggs and Cox studied the restricted online Ramsey numbers of matchings and determined a general upper bound for them. They proved that for $n=3r-1=R_2(r K_2)$ we have $\tilde{R}_{2}(r K_2;n) \leq n-1$ and asked whether this was tight. In this short note, we provide a general lower bound for these Ramsey numbers. As a corollary, we answer this question of Briggs and Cox, and confirm that for $n=3r-1$ we have $\tilde{R}_{2}(r K_2;n) = n-1$. We also show that for $n'=4r-2=R_3(r K_2)$ we have $\tilde{R}_{3}(r K_2;n') = 5r-4$.

The size-Ramsey number of 3-uniform tight paths

Given a hypergraph H, the size-Ramsey number r(H) is the smallest integer m such that there exists a graph G with m edges with the property that in any colouring of the edges of G with two colours there is amonochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices P_n is linear in n, i.e., r(P_n)=O(n). This answers a question by Dudek, Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved r(P_n)=O(n^1.5*log^1.5 n).