arXiv subjects

Krzysztof Choromanski

Publications and source records attributed to Krzysztof Choromanski.

111 records · Page 7Linked to original sources

arXiv2014

Excluding pairs of tournaments

The Erdős-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.

math.CO↗

arXiv2014

All known prime Erdős-Hajnal tournaments satisfy $ε(H) = Ω(\frac{1}{|H|^{5}\log(|H|)})$

We prove that there exists $C>0$ such that $ε(H) \geq \frac{C}{|H|^{5}\log(|H|)}$, where $ε(H)$ is the Erdős-Hajnal coefficient of the tournament $H$, for every prime tournament $H$ for which the celebrated Erdős-Hajnal Conjecture has been proven so far. This is the first polynomial bound on the EH coefficient obtained for all known prime Erdős-Hajnal tournaments, in particular for infinitely many prime tournaments. As a byproduct of our analysis, we answer affirmatively the question whether there exists an infinite family of prime tournaments $H$ with $ε(H)$ lower-bounded by $\frac{1}{\textit{poly}(|H|)}$, where $\textit{poly}$ is a polynomial function. Furthermore, we give much tighter bounds than those known so far for the EH coefficients of tournaments without large homogeneous sets. This enables us to significantly reduce the gap between best known lower and upper bounds for the EH coefficients of tournaments. As a corollary we prove that every known prime Erdős-Hajnal tournament $H$ satisfies: $-5 + o(1) \leq \frac{\log(ε(H))}{\log(|H|)} \leq -1 + o(1)$. No lower bound on that expression was known before. We also show the applications of those results to the tournament coloring problem. In particular, we prove that for every known prime Erdős-Hajnal tournament $H$ every $H$-free tournament has \textit{chromatic number} at most $O(n^{1-\frac{C}{|H|^{5}\log(|H|)}}\log(n))$, where $C>0$ is some universal constant. The related coloring can be constructed algorithmically in the quasipolynomial time by following straightforwadly the proof of our main result. In comparison, the standard Ramsey theory gives only $O(\frac{n}{\log(n)})$ bounds for the tournament chromatic number.

math.CO↗

arXiv2014

The Strong EH-Property and the Erdős-Hajnal Conjecture

The Erdős-Hajnal Conjecture states that for every $H$ there exists a constant $ε(H)>0$ 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 $|V(G)|^{ε(H)}$. The Conjecture is still open. Some time ago its directed version was formulated (see:\cite{alon}). In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. If the Conjecture is not true then the smallest counterexample is a prime tournament. For a long time the Conjecture was known only for finitely many prime tournaments. Recently in \cite{bcc} and \cite{choromanski2} the Conjecture was proven for the families of galaxies and constellations that contain infinitely many prime tournaments. In \cite{bcc} the Conjecture was also proven for all $5$-vertex tournaments. We say that a tournament $H$ has the $EH$-property if it satisfies the Conjecture. In this paper we introduce the so-called \textit{strong EH-property} which enables us to prove the Conjecture for new prime tournaments, but what is even more interesting, provides a mechanism to combine tournaments satisfying the Conjecture to get bigger tournaments that do so and are not necessarily nonprime. We give several examples of families of tournaments constructed according to this procedure. The only procedure known before used to construct bigger tournaments satisfying the Conjecture from smaller tournaments satisfying the Conjecture was the so-called \textit{substitution procedure} (see: \cite{alon}). However an outcome of this procedure is always a nonprime tournament and, from what we have said before, prime tournaments are those that play crucial role in the research on the Conjecture. Our method may be potentially used to prove the Conjecture for several new classes of tournaments.

math.CO↗