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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.

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique and its algorithmic version, enhanced with a roll-back idea allowing a sequential retracing of previously performed embedding steps. We use this method to obtain the following results. We show that the size-Ramsey number of logarithmically long subdivisions of bounded degree graphs is linear in their number of vertices, settling a conjecture of Pak [Proceedings of the thirteenth annual ACMSIAM symposium on discrete algorithms (SODA’02), 2002, pp. 321-328]. We give a deterministic, polynomial time online algorithm for finding vertex-disjoint paths of a prescribed length between given pairs of vertices in an expander graph. Our result answers a question of Alon and Capalbo [48th annual IEEE symposium on foundations of computer science (FOCS’07), 2007, pp. 518-524]. We show that relatively weak bounds on the spectral ratio λ / d \lambda /d of d d -regular graphs force the existence of a topological minor of K t K_t where t = ( 1 − o ( 1 ) ) d t=(1-o(1))d . We also exhibit a construction which shows that the theoretical maximum t = d + 1 t=d+1 cannot be attained even if λ = O ( d ) \lambda =O(\sqrt {d}) . This answers a question of Fountoulakis, Kühn and Osthus [Random Structures Algorithms 35 (2009), pp. 444-463].

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems:• For a graph H, we denote by H q the graph obtained from H by subdividing its edges with q−1 vertices each. We show that the k-size-Ramsey number Rk (H q ) satisfies Rk (H q ) = O(qn) for every bounded degree graph H on n vertices and for q = Ω(log n), which is optimal up to a constant factor. This settles a conjecture of Pak ( 2002). • We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between given pairs of vertices in a strong expander graph. More precisely, let G be an (n, d, λ)-graph with λ = O(d 1−ε ), and let P be any collection of at most c n log d log n disjoint pairs of vertices in G for some small constant c, such that in the neighborhood of every vertex in G there are at most d/4 vertices from P. Then there exists a polynomial time algorithm which finds vertex-disjoint paths between every pair in P, and each path is of the same length = O log n log d . Both the number of pairs and the length of the paths are optimal up to a constant factor; the result answers the offline version of a question of Alon and Capalbo (2007).

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result from the 1980's claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of Solecki and Zhao that the class of all finite posets with several linear extensions has the Ramsey property.Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property.