Constructive Ramsey Numbers for Loose Hyperpaths

@inproceedings{Dudek2018ConstructiveRN,
  title={Constructive Ramsey Numbers for Loose Hyperpaths},
  author={Andrzej Dudek and Andrzej Rucinski},
  booktitle={LATIN},
  year={2018}
}
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… 
Monochromatic loose paths in multicolored k-uniform cliques
TLDR
There is an algorithm such that for every $r$-edge-coloring of the edges of the complete $k$-uniform hypergraph, it finds a monochromatic copy of P_\ell^{(k)}$ in time at most $cn^k$.

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