3 problems

Maximum-degree phase-transition conjecture. There is a threshold $\alpha_1$ on the average degree such that below it an optimal graph has a vertex of degree $\Omega(n)$, whereas ab…

Leaf-density phase-transition conjecture. There is a threshold $\alpha_0$ on the average degree such that below it all optimal graphs have a positive proportion of leaves, whereas…

Convex lower-bound conjecture. For all $x$,