7 problems

Let $f^*(2,3,n)$ denote the random variable measuring the largest number of vertices that can be reached from a vertex by monochromatic directed paths of depth at most $3$ in a two…

Consider a random tournament on $n$ vertices, and color each edge with one of two colors. A vertex monochromatically dominates another vertex using paths of depth at most $3$ if th…

Let $p_{\min}:=\min_{i\ne j}p_{i,j}$ denote the minimum probability of an edge between distinct alternatives in the generalized random tournament model. A 5-kings threshold conject…

Equal-likelihood conjecture. For any fixed $k$, all tournaments on $k$ dice are equally likely.

Let $k$ random dice determine a tournament on $k$ alternatives by orienting each pair according to the beats relation: an edge points from die $\mathbf a$ to die $\mathbf b$ when…

Let $T$ be a random tournament on $n$ vertices, and let $m(T)$ denote the maximum number of vertices in a monochromatic path in a $2$-colouring of the edges of $T$, minimized over…

Degree-two vertex conjecture. If $s$ has degree at most two, then the events $\{a\to s\}$ and $\{s\to b\}$ are independent or negatively correlated.