5 problems

Let $G$ be the annealed randomly oriented Erdős–Rényi graph $\vec G(n,p)$, with distinct vertices $a,s,b,t$, and let $\{s\to a\}$ and $\{t\to b\}$ be the events that directed paths…

Single-sign-change conjecture. For any fixed $n\ge 5$, the covariance in $G(n,m)$ changes sign only once between two values of $m$.

Half-probability positivity conjecture. For $p=\frac12$ and $n\ge 6$, the relative covariance $R_n(p)$ is positive.

Let $p_1(n)<p_2(n)<p_3(n)$ be the three critical probabilities from the preceding conjecture for the relative covariance in the randomly oriented graph $G(n,p)$. Asymptotic sign-ch…

The three-sign-change conjecture. For $n\ge 27$, the relative covariance $R_n(p)$ changes sign at three critical probabilities $p_1(n)<p_2(n)<p_3(n)$.