• DOI:10.23638/DMTCS-21-4-7
• Corpus ID: 56227552

Monochromatic loose paths in multicolored k-uniform cliques

@article{Dudek2019MonochromaticLP,
  title={Monochromatic loose paths in multicolored k-uniform cliques},
  author={Andrzej Dudek and Andrzej Rucinski},
  journal={Discret. Math. Theor. Comput. Sci.},
  year={2019},
  volume={21}
}

• A. Dudek, A. Rucinski
• Published 13 March 2018
• Mathematics
• Discret. Math. Theor. Comput. Sci.

For positive integers $k$ and $\ell$, a $k$-uniform hypergraph is called a \emph{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 \emph{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 $R(P_\ell^{(k)};r)\le(k-1)\ell r$.