Constructive Ramsey numbers for loose hyperpaths

Abstract

For positive integers k and ℓ, a k-uniform hypergraph is called a loose path of length ℓ, and denoted by Pℓ(k), if its vertex set is {v1, v2, …, v(k-1)ℓ+1} and the edge set is {ei={v(i-1)(k-1)+q:1≤q≤k},i=1,⋯,ℓ}, that is, each pair of consecutive edges intersects on a single vertex. Let R(Pℓ(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 Kn(k) yields a monochromatic copy of Pℓ(k). In this note we are interested in constructive upper bounds on R(Pℓ(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 Pℓ(k) in every coloring. In particular, we show that there is a constant c> 0 such that for all k≥ 2, ℓ≥ 3, 2 ≤ r≤ k- 1, and n≥ k(ℓ+ 1) r(1 + ln (r)), there is an algorithm such that for every r-edge-coloring of the edges of Kn(k), it finds a monochromatic copy of Pℓ(k) in time at most cnk.