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

Density version of the Ramsey problem and the directed Ramsey problem

We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges $|E_{RB}|$ is given. The aim is to find the maximal size $f$ of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on $n$ vertices such that the number of bi-oriented edges $|E_{bi}|$ is given. The aim is to bound the size $F$ of the maximal transitive subtournament that is guaranteed by such a digraph. Applying probabilistic and analytic tools and constructive methods we show that if $|E_{RB}|=|E_{bi}| = p{n\choose 2}$, ($p\in [0,1)$), then $f, F < C_p\log(n)$ where $C_p$ only depend on $p$, while if $m={n \choose 2} - |E_{RB}| <n^{3/2}$ then $f= \Theta (\frac{n^2}{m+n})$. The latter case is strongly connected to Tur\'an-type extremal graph theory.Comment: 17 pages. Further lower bound added in case $|E_{RB}|=|E_{bi}| = p{n\choose 2}

On the minimum degree of minimal Ramsey graphs for multiple colours

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) = r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r

Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201