Finite palette endpoints and degree-square Turán problems

We study finite palette extremal problems motivated by uniform Turán densities of $3$-uniform hypergraphs. Given a self-converse tournament $T$ with at least two vertices, we determine the largest possible number of admissible triples in an $m$-color palette that avoids the left and right palettes associated with $T$. The answer is the one-sided degree-square Turán number $${\mathrm{pal}}_T(m)={\mathrm{ex}}_2^{+}(m,T)=\max \{\sum _{v\in V(D)}{d}_D^{+}(v{)}^2:\mathrm{\mid }V(D)\mathrm{\mid }=m,D\text{ is }T\text{-free}\}\mathrm{.}$$ Thus this palette problem is reduced to an extremal problem for digraph out-degrees. We then prove a prefix-majorization lemma for convex out-degree moments and apply it to directed cycles. In particular, ${\mathrm{ex}}_2^{+}(m,{\overrightarrow{C}}_3)=\frac{m(m^2-1)}{3}$, which gives the sharp $m$-color palette endpoint $\frac{1}{3}-\frac{1}{3m^2}$ for the directed triangle. Combining this endpoint with the palette characterization of uniform Turán density and the palette separation theorem, we show that for every $m\ge 2$ there is a finite $3$-graph $H_m$ such that $$\frac{1}{3}-\frac{1}{3m^2}\le {\pi }_u(H_m)\le \frac{1}{3}\mathrm{.}$$ Hence there is a sequence of individual finite $3$-graphs whose uniform Turán densities converge to $1\mathrm{/}3$. We also describe the extremal palettes, prove a qualitative edit-distance stability theorem, and compute the Lagrangian of the endpoint palette ${\mathcal{P}}_m^{\star }$. As a consequence, for every $m\ge 2$ there is a finite family ${\mathcal{F}}_m$ of $3$-graphs with ${\pi }_u({\mathcal{F}}_m)=\frac{1}{3}-\frac{1}{3m^2}$, so $1\mathrm{/}3$ is an accumulation point of uniform Turán densities of finite forbidden families.