Excluding pairs of tournaments

The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph $H$ there exists a constant $c(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $|V(G)|^{c(H)}$. The conjecture is still open. Its equivalent directed version states that for every given tournament $H$ there exists a constant $c(H)>0$ such that every $H$-free tournament $T$ contains a transitive subtournament of order at least $|V(T)|^{c(H)}$. We prove in this paper that $\{H_{1},H_{2}\}$-free tournaments $T$ contain transitive subtournaments of size at least $|V(T)|^{c(H_{1},H_{2})}$ for some $c(H_{1},H_{2})>0$ and several pairs of tournaments: $H_{1}$, $H_{2}$. In particular we prove that $\{H,H^{c}\}$-freeness implies existence of the polynomial-size transitive subtournaments for several tournaments $H$ for which the conjecture is still open ($H^{c}$ stands for the \textit{complement of $H$}). To the best of our knowledge these are first nontrivial results of this type

Dominating the Erdos-Moser theorem in reverse mathematics

The Erdos-Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (omega-model) of simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner.Comment: 36 page

Structure theorem for U5-free tournaments

Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2 \rightarrow v_1$, and $v_i \rightarrow v_j$ if $j-i \equiv 1$, $2 \pmod{5}$ and ${i,j} \neq {1,2}$. In this paper we describe the tournaments which do not have $U_5$ as a subtournament. Specifically, we show that if a tournament $G$ is "prime"---that is, if there is no subset $X \subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v \in V(G) \backslash X$, either $v \rightarrow x$ for all $x \in X$ or $x \rightarrow v$ for all $x \in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$ or $V(G)$ can be partitioned into sets $X$, $Y$, $Z$ such that $X \cup Y$, $Y \cup Z$, and $Z \cup X$ are transitive. From the prime $U_5$-free tournaments we can construct all the $U_5$-free tournaments. We use the theorem to show that every $U_5$-free tournament with $n$ vertices has a transitive subtournament with at least $n^{\log_3 2}$ vertices, and that this bound is tight.Comment: 15 pages, 1 figure. Changes from previous version: Added a section; added the definitions of v, A, and B to the main proof; general edit

No additional tournaments are quasirandom-forcing

A tournament H is quasirandom-forcing if the following holds for every sequence (G_n) of tournaments of growing orders: if the density of H in G_n converges to the expected density of H in a random tournament, then (G_n) is quasirandom. Every transitive tournament with at least 4 vertices is quasirandom-forcing, and Coregliano et al. [Electron. J. Combin. 26 (2019), P1.44] showed that there is also a non-transitive 5-vertex tournament with the property. We show that no additional tournament has this property. This extends the result of Bucic et al. [arXiv:1910.09936] that the non-transitive tournaments with seven or more vertices do not have this property

Some results and problems on tournament structure

This paper is a survey of results and problems related to the following question: is it true that if G is a tournament with sufficiently large chromatic number, then G has two vertex-disjoint subtournaments A,B, both with large chromatic number, such that all edges between them are directed from A to B? We describe what we know about this question, and report some progress on several other related questions, on tournament colouring and domination