35 problems

Alspach–Mason–Pullman conjecture. This lower bound is attained for every tournament $T$ of even order.

Let $D$ be a set of nonnegative integers, and let a tournament $T$ be an orientation of a complete graph; its degree set is the set of distinct out-degrees of its vertices. Reid's…

Let $T$ be a tournament and let $R\subseteq V(T)$ be a minimal TEQ-retentive set with $|R|=3$. Size-three uniqueness conjecture. Then $R$ is the unique minimal TEQ-retentive set of…

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…

An oriented graph is a graph whose edges are assigned directions. An oriented graph is tournament anti-Sidorenko when its homomorphism density in every tournament is at most…

An oriented path is a path whose edges have been assigned directions; let $k$ denote its number of arcs. An oriented path is tournament anti-Sidorenko when its homomorphism density…

An oriented path is a path whose edges have been assigned directions, and a block is a maximal subpath whose arcs all have the same direction. An oriented graph is tournament anti-…

Let $W_u$ be a Walecki tournament with vertex labelled $*$. An automorphism of $W_u$ is an isomorphism from $W_u$ to itself, and an isomorphism between two Walecki tournaments is a…

An $r$-round deterministic ordered balanced knockout tournament is an ordered balanced knockout tournament whose format is deterministic, and an $r$-round dramatic ordered balanced…

A totally randomized knockout tournament is a knockout bracket in which teams are randomly placed onto the starting lines rather than placed according to seed. Chung and Hwang's co…

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…

Conjecture 3. The assertion cannot be reconstructed from the supplied text.

Non-critical vertex conjecture. For every $\ell$ with $n-d+3\le \ell\le d$, there exists a non-critical vertex $w$ such that the diameter of the strongly connected subtournament…

Savchenko's Moon-type conjecture. For every $3\le \ell\le n$,

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…

No-triple-dominance conjecture. No triple is dominant in ${\cal T}_4$.

Constant-density conjecture. For all $h$, this fraction is at least a positive constant independent of $h$.

Entropy conjecture. For $\alpha$ sufficiently large,

Bradley–Terry density conjecture. The closure in $\mathbb{R}^n$ of the set $\mathcal{X}_n^{B\text{-}T}$ is $\mathcal{X}_n$. Although the Bradley–Terry model appears more restrictiv…

Rainbow-triangle bound conjecture. For the existence of rainbow triangles in arc-colored regular tournaments, the bound

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…

Balanced-format conjecture. Among all possible formats, the balanced format maximizes, and the format with exactly one match in each round minimizes, the best player's probability…

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…