Bollobás–Przykucki–Riordan–Sahasrabudhe conjecture for the $K_5$ running time

Let $M_{K_5}(n)$ denote the maximum running time of the $K_5$ infection process over starting graphs on $n$ vertices. Bollobás–Przykucki–Riordan–Sahasrabudhe conjecture.

$$M_{K_5}(n)=o(n^2).$$

The conjecture concerns the outstanding determination of the asymptotic maximum running time for the $K_5$ process. The paper notes that the conjectured upper bound was proved by Balogh et al. using a construction based on Behrend sets.