Irredundant sets, Ramsey numbers, multicolor Ramsey numbers

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ that is adjacent to $x$ and to no other vertex of $X$. The \emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in a blue subgraph or a $n$-element independent set in a red subgraph. The \emph{multicolor irredundant Ramsey number} $s(t_{1},\ldots,t_{l})$ is the minimum $r$ such that every $l$-coloring of the edges of the complete graph $K_{r}$ on $r$ vertices has a monochromatic irredundant set of size $s_{i}$ for certain $1\leq i\leq l$. Firstly, we improve the upper bound for the mixed Ramsey number $t(3,n)$, and using this result, we verify a special case of a conjecture proposed by Chen, Hattingh, and Rousseau for $m=4$. Secondly, we obtain a new upper bound for $s(3,9)$, and using Krivelevich's method, we establish an asymptotic lower bound for CO-irredundant Ramsey number of $K_{N}$, which extends Krivelevich's result on $s(m,n)$. Thirdly, we prove a lower bound for the multicolor irredundant Ramsey number by a random and probability method which has been used to improve the lower bound for multicolor Ramsey numbers. Finally, we give a lower bound for the irredundant multiplicity.Comment: 23 pages, 1 figur

Improved bounds on the multicolor Ramsey numbers of paths and even cycles

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest

Monochromatic loose paths in multicolored kk-uniform cliques

For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of $\ell$ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap e_j=\emptyset$ if $|i-j|\ge2$. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let $R(P_\ell^{(k)};r)$ be the minimum integer $n$ such that every $r$-edge-coloring of the complete $k$-uniform hypergraph $K_n^{(k)}$ yields a monochromatic copy of $P_\ell^{(k)}$. In this paper we are mostly interested in constructive upper bounds on $R(P_\ell^{(k)};r)$, meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of $P_\ell^{(k)}$ in every coloring. In particular, we show that there is a constant $c>0$ such that for all $k\ge 2$, $\ell\ge3$, $2\le r\le k-1$, and $n\ge k(\ell+1)r(1+\ln(r))$, there is an algorithm such that for every $r$-edge-coloring of the edges of $K_n^{(k)}$, it finds a monochromatic copy of $P_\ell^{(k)}$ in time at most $cn^k$. We also prove a non-constructive upper bound $R(P_\ell^{(k)};r)\le(k-1)\ell r$