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 $R(P_\ell^{(k)};r)\le(k-1)\ell r$

On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page

On globally sparse Ramsey graphs

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available