Tournaments With Forbidden Substructures and the Erdos-Hajnal Conjecture
type: CreativeWork
description: A celebrated Conjecture of Erdos and Hajnal states that for every undirected graph H there exists ɛ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n^{ɛ(H)}. In 2001 Alon, Pach and Solymosi proved that the conjecture has an equivalent directed version, where undirected graphs are replaced by tournaments and cliques and stable sets by transitive subtournaments. This dissertation addresses the directed version of the conjecture and some problems in the directed setting that are closely related to it. For a long time the conjecture was known to be true only for very specific small graphs and graphs obtained from them by the so-called substitution procedure proposed by Alon, Pach and Solymosi. All the graphs that are an outcome of this procedure have nontrivial homogeneous sets. Tournaments without nontrivial homogeneous sets are called prime. They play a central role here since if the conjecture is not true then the smallest counterexample is prime. We remark that for a long time the conjecture was known to be true only for some prime graphs of order at most 5. There exist 5-vertex graphs for which the conjecture is still open, however one of the corollaries of the results presented in the thesis states that all tournaments on at most 5 vertices satisfy the conjecture. In the first part of the thesis we will establish the conjecture for new infinite classes of tournaments containing infinitely many prime tournaments. We will first prove the conjecture for so-called constellations. It turns out that almost all tournaments on at most 5 vertices are either constellations or are obtained from constellations by substitutions. The only 5-vertex tournament for which this is not the case is a tournament in which every vertex has outdegree 2. We call this the tournament C_{5}. Another result of this thesis is the proof of the conjecture for this tournament. We also present here…
abstract: A celebrated Conjecture of Erdos and Hajnal states that for every undirected graph H there exists ɛ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n^{ɛ(H)}. In 2001 Alon, Pach and Solymosi proved that the conjecture has an equivalent directed version, where undirected graphs are replaced by tournaments and cliques and stable sets by transitive subtournaments. This dissertation addresses the directed version of the conjecture and some problems in the directed setting that are closely related to it. For a long time the conjecture was known to be true only for very specific small graphs and graphs obtained from them by the so-called substitution procedure proposed by Alon, Pach and Solymosi. All the graphs that are an outcome of this procedure have nontrivial homogeneous sets. Tournaments without nontrivial homogeneous sets are called prime. They play a central role here since if the conjecture is not true then the smallest counterexample is prime. We remark that for a long time the conjecture was known to be true only for some prime graphs of order at most 5. There exist 5-vertex graphs for which the conjecture is still open, however one of the corollaries of the results presented in the thesis states that all tournaments on at most 5 vertices satisfy the conjecture. In the first part of the thesis we will establish the conjecture for new infinite classes of tournaments containing infinitely many prime tournaments. We will first prove the conjecture for so-called constellations. It turns out that almost all tournaments on at most 5 vertices are either constellations or are obtained from constellations by substitutions. The only 5-vertex tournament for which this is not the case is a tournament in which every vertex has outdegree 2. We call this the tournament C_{5}. Another result of this thesis is the proof of the conjecture for this tournament. We also present here…
author: Choromanski, Krzysztof
language: English
keywords: Mathematics
date published: 2013
date modified: 2026-07-24T01:44:27Z
provider: Columbia University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Columbia University

PropertyValue
property ID: DOI
value: 10.7916/D84J0N9P