Linear vertex threshold conjecture for isolated cliques in the minimum-degree Kruskal–Katona problem

Let $t$ be the clique parameter, let $n$ be the number of vertices, and consider Problem 13 with $k=3$. An extremal graph is a graph attaining the extremal value in that problem, and an isolated copy of $K_{t+1}$ is a copy with no edges joining its vertices to the rest of the graph.

Linear threshold conjecture. If

$$n \geq Ct$$

for some absolute constant $C$, then every extremal graph for Problem 13 with $k=3$ contains an isolated copy of $K_{t+1}$.

The theorem established in the paper proves the analogous structural property under the quadratic bound $n\geq ct^2+o(t^2)$, with $c=1+\sqrt{928/33}$. This conjecture asks whether the threshold can be reduced to a linear bound in $t$.