Tight paths in fully directed hypergraphs

It is well-known that every tournament has a spanning path. We consider hypergraph analogues. In an r-uniform fully directed hypergraph , or r-digraph , every edge is a list of r distinct vertices. An ( 𝑟 , 𝑘 ) -tournament is an r -digraph G such that for every r -set S of vertices in G , exactly k of the orderings of S are edges in G . A directed tight path is an r -digraph G whose vertices can be ordered so that the intervals of size r are the edges in G . Let 𝑓 ⁡ ( 𝑛 , 𝑟 , 𝑘 ) be the maximum s such that every n -vertex ( 𝑟 , 𝑘 ) -tournament contains a tight path on s vertices. Since every tournament has a spanning path, we have 𝑓 ⁡ ( 𝑛 , 2 , 1 ) = 𝑛 . In this paper, we show that the minimum k such that 𝑓 ⁡ ( 𝑛 , 𝑟 , 𝑘 ) tends to infinity with n is in the interval [ ( 1 − 1 𝑟 − 𝑂 ⁡ ( l o g ⁡ 𝑟 𝑟 2 ⁢ l o g ⁡ l o g ⁡ 𝑟 ) ) ⁢ 𝑟 ! , ( 1 − 1 𝑟 − 𝜑 ⁡ ( 𝑟 ) − 1 𝑟 ! ) ⁢ 𝑟 ! ] where 𝜑 ⁡ ( 𝑟 ) is the Euler Totient Function, and we find the exact value when 𝑟 ≤ 5 . We also show that Ω ⁡ ( √ l o g ⁡ 𝑛 / ⁢ l o g ⁡ l o g ⁡ 𝑛 ) ≤ 𝑓 ⁡ ( 𝑛 , 3 , 3 ) ≤ 𝑂 ⁡ ( l o g ⁡ 𝑛 ) and 𝑓 ⁡ ( 𝑛 , 3 , 4 ) ≥ Ω ⁡ ( 𝑛 1 / 5 ) .