5 problems

Peng–Zhao's conjecture. The Lagrangian of $G$ satisfies

Let $r\geq 3$. An $r$-graph is linear if any two edges have at most one vertex in common, and it is lambda-perfect when its Lagrangian density equals the natural complete-hypergrap…

Lagrangian-density conjecture. The Lagrangian density of $K_4^3\cup e$ is attained by $S_2(n)$ as $n\to\infty$. The conjecture is proposed as an approach to the unresolved Turán-de…

Let $r\geq 3$. An $r$-graph $G$ has $m$ edges and is lambda-perfect when its Lagrangian density equals the natural complete-hypergraph lower bound determined by its number of verti…

Let $r$ be a positive integer and let $m$ be the number of edges. For an $r$-graph, its Lagrangian is the maximum of its edge-weight polynomial over all nonnegative vertex weights…