12 problems

A tournament $H$ is a champion if there exists an integer $c_H$ such that every $H$-free tournament has acyclic dichromatic number at most $c_H$. The notation $TT_k$ denotes the tr…

Let $T$ be an $n$-vertex tournament. An inversion reverses all edges whose endpoints lie in a specified vertex set, and ${\rm inv}_k(T)$ is the minimum length of a sequence of inve…

Let $T_k$ be the transitive tournament on $k$ vertices, and let $H_k$ be its $1$-subdivision, obtained by subdividing every arc exactly once. For an oriented graph $H$, let…

Let $C_k$ be the $k$-vertex carousel tournament, and let $C_n$ be the carousel tournament on $n$ vertices. For a tournament $T$, write $I(C_k,T)$ for the number of induced copies o…

Let $T$ be a tournament. A nebula is a tournament admitting an ordering whose vertices are partitioned into the vertex sets of stars and singleton components, with no restriction o…

Let $D$ be a directed graph with vertex set $V(D)$ and edge set $E(D)$. A path decomposition of $D$ is a collection of directed paths whose edge sets partition $E(D)$. Let…

Entropy conjecture. For $\alpha$ sufficiently large,

Let $\mathcal{A}_n$ denote the set of score vectors attainable in tournaments on $n$ players, and let $\mathcal{A}^{*}_n$ denote the region generated by the tournament configuratio…

Let $T$ be a tournament. A captain vertex is a vertex with the captain property used in the paper, and let $A$ be the set of all captain vertices of $T$. Captain inclusion conjectu…

Let $T$ be a tournament, and let $\mathfrak{Dom}(T)$ denote its directed domination graph. A set is TEQ-retentive when it satisfies the tournament-equilibrium-set retentiveness con…

Carousel maximizer uniqueness conjecture. For every $n\geq2$, a homomorphism $\phi\in\operatorname{Hom}^+({\mathcal{A}}^0,\mathbb{R})$ maximizes the density of $R_{2n+1}$ if and on…

Carousel homomorphism conjecture. For every $n\in\mathbb{N}$, the carousel homomorphism maximizes the density of $R_{2n+1}$: