On 1-subdivisions of transitive tournaments

Abstract

The oriented Ramsey number r(H) for an acyclic digraph H is the minimum integer n such that any n-vertex tournament contains a copy of H as a subgraph. We prove that the 1-subdivision of the k-vertex transitive tournament H k satisfies r(H k ) ≤ O(k 2 log log k). This is tight up to multiplicative log log k-term.We also show that if T is an n-vertex tournament with ∆ + (T )arcs. This is also tight up to multiplicative constant.2 for an n-vertex graph G to contains H as a subgraph was determined by Erdős-Stone [7] and . This threshold is quadratic in n. However, if we only want to find such a structure as a subdivision rather than as a subgraph, much weaker bound is sufficient.In 1967, Mader [12] proved that for given k, there exists f (k) such that every graph with average degree f (k) contains K k as a subdivision. Mader [12] and Erdős-Hajnal [4] conjectured that this f (k) can be shown to be O(k 2 ) and this was verified by Bollobás and Thomason [3] and independently by Komlós and Szemerédi [10].Another key question in extremal combinatorics is a Ramsey-type question. For a given H, what values of n ensure that any 2-coloring on the edges of K n contains a monochromatic H? We write r(H) to denote the smallest such n. In general, such a number r(H) is exponential in |H| as it is well-known that Ω(2 k ) ≤ r(K k ) ≤ 4 k−O(log 2 k) . However, Alon [1] in 1994 proved that if H is a subdivision of another graph, the Ramsey number r(H) is linear in |H|. Note that such a graph H is always 2-degenerate. This result was further improved by the celebrated result of Lee [11] in 2017 proving the Burr-Erdős conjecture stating that any d-degenerate graph has linear Ramsey number.There is an analogue considering tournaments instead of complete graphs. For a given oriented graph H, we define the oriented Ramsey number r(H) to be the smallest n where any n-vertex tournament contains a copy of H. As a transitive tournament on n vertices, which we denote by T n , does not contain any digraph with a cycle as a subgraph, r(H) is only defined for acyclic digraphs H. More generally, for a collection H of oriented graphs, we define r(H) to be the smallest n where any n-vertex tournament contains a copy of a graph in H. Again, at least one graph in H has to be acyclic for the parameter to be defined.Stearns [13] in 1959 and Erdős and Moser [5] in 1964 initiated the study on the oriented Ramsey number and proved that 2 k/2−1 ≤ r(T k ) ≤ 2 k−1 .