Powers of paths in tournaments - Draganić, Nemanja; Dross, François; Fox, Jacob; Girao, Antonio; Havet, Frédéric; Korándi, Dániel; Lochet, William; Correia, David Munhá; Scott, Alex; Sudakov, Benny (Combinatorics, probability & computing, 2021)
In this short note we prove that every tournament contains the k-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when k=2, showing that there is always a square of a directed path of length , which is best possible.