Related papers: Dominant tournament families
A tournament $H$ is said to force quasirandomness if it has the property that a sequence $(T_n)_{n\in \mathbb{N}}$ of tournaments of increasing orders is quasirandom if and only if the homomorphism density of $H$ in $T_n$ tends to…
We study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a…
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.…
A 3-tournament is a complete 3-uniform hypergraph where each edge has a special vertex designated as its tail. A vertex set $X$ dominates $T$ if every vertex not in $X$ is contained in an edge whose tail is in $X$. The domination number of…
A celebrated unresolved conjecture of Erd\H{o}s and Hajnal states that for every undirected graph $H$ there exists $\epsilon(H)>0$ such that every undirected graph on $n$ vertices that does not contain $H$ as an induced subgraph contains a…
A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of…
In this thesis we prove a variety of theorems on tournaments. A \emph{prime} tournament is a tournament $G$ such that there is no $X \subseteq V(G)$, $1 < |X| < |V(G)|$, such that for every vertex $v \in V(G) \minus X$, either $v \ra x$ for…
A pure pair in a tournament $G$ is an ordered pair $(A,B)$ of disjoint subsets of $V(G)$ such that every vertex in $B$ is adjacent from every vertex in $A$. Which tournaments $H$ have the property that if $G$ is a tournament not containing…
A $k$-tournament $H$ on $n$ vertices is a pair $(V, A)$ for $2\leq k\leq n$, where $V(H)$ is a set of vertices, and $A(H)$ is a set of all possible $k$-tuples of vertices, such that for any $k$-subset $S$ of $V$, $A(H)$ contains exactly one…
The Erd\H{o}s-Hajnal Conjecture states that for every $H$ there exists a constant $\epsilon(H)>0$ 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…
Suppose one needs to change the direction of at least $\epsilon n^2$ edges of an $n$-vertex tournament $T$, in order to make it $H$-free. A standard application of the regularity method shows that in this case $T$ contains at least…
A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it…
A celebrated unresolved conjecture of Erd\"{o}s and Hajnal states that for every undirected graph $H$ there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph…
A well-known theorem of Chung and Graham states that if $h\geq 4$ then a tournament $T$ is quasirandom if and only if $T$ contains each $h$-vertex tournament the "correct number" of times as a subtournament. In this paper we investigate the…
A (simple) hypergraph is a family H of pairwise incomparable sets of a finite set. We say that a hypergraph H is a domination hypergraph if there is at least a graph G such that the collection of minimal dominating sets of G is equal to H.…
The R\'{e}nyi $\alpha$-entropy $H_{\alpha}$ of complete antisymmetric directed graphs (i.e., tournaments) is explored. We optimize $H_{\alpha}$ when $\alpha = 2$ and $3$, and find that as $\alpha$ increases $H_{\alpha}$'s sensitivity to…
An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon, Pack, and Solymosi states that for every tournament H there exists epsilon(H)>0 such that every H-free n-vertex tournament T…
A celebrated unresolved conjecture of Erd\"{o}s and Hajnal states that for every undirected graph $H$ there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph…
We give a complete characterization of tournaments H that have the Sidorenko property with respect to nearly regular tournaments, i.e., the homomorphism density of H among all nearly regular tournaments is minimized by a random tournament.…
We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Haddad, Pouzet and Sch\"olzel [1]. We observe that on a set $V$ with $m$ elements, there is a hereditarily rigid set $\mathcal R$ made of $n$…