Extremal partition conjecture for generalized Ramsey–Turán clique density

Let $s$ and $t$ be integers with $3\leq s\leq t-2$. An extremal graph for the generalized Ramsey–Turán density $\varrho_s(K_t)$ is a graph admitting either an $(s,t-1-s)$-partition when $s+2\leq t\leq 2s-1$, or a $(\lfloor t/2\rfloor,t-1-\lfloor t/2\rfloor)$-partition when $t\geq 2s$. Extremal partition conjecture. There is an extremal graph for $\varrho_s(K_t)$ with the corresponding partition. The conjecture proposes a periodic extremal structure for generalized Ramsey–Turán densities; the paper studies conditions under which this structure holds and gives counterexamples, so its resolution is not established in the supplied text.