LATIN 2018: LATIN 2018: Theoretical Informatics pp 427-436

# Constructive Ramsey Numbers for Loose Hyperpaths

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

## Abstract

For positive integers k and $$\ell$$, a k-uniform hypergraph is called a loose path of length $$\ell$$, and denoted by $$P_\ell ^{(k)}$$, if its vertex set is $$\{v_1, v_2, \ldots , v_{(k-1)\ell +1}\}$$ and the edge set is $$\{e_i = \{ v_{(i-1)(k-1)+q} : 1 \le q \le k \},\ i=1,\dots ,\ell \}$$, that is, each pair of consecutive edges intersects on a single vertex. Let $$R(P_\ell ^{(k)};r)$$ be the multicolor Ramsey number of a loose path that is the minimum 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 note we are interested in constructive upper bounds on $$R(P_\ell ^{(k)};r)$$ which means 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 $$k\ge 2$$, $$\ell \ge 3$$, $$2\le r\le k-1$$, 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 $$cn^k$$.

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