Finite palette endpoints and degree-square Tur\'an problems

We study finite extremal problems for palettes, which arise from the palette framework for the uniform Tur\'an densities of $3$-uniform hypergraphs. Recent work has developed reductions from palette colorability questions to extremal problems for digraphs. In this paper we prove an exact degree-square refinement of these reductions for a natural family of left and right tournament palettes. For a tournament $T$, let $P_T^L$ and $P_T^R$ denote the left and right palettes generated by $T$. We prove that if $T$ is self-converse and has at least two vertices, then for every $m\ge 1$ the maximum number of admissible triples in an $m$-color palette avoiding both $P_T^L$ and $P_T^R$ is \[ \operatorname{ex}_2^+(m,T) = \max\left\{ \sum_{v\in V(D)} d_D^+(v)^2: |V(D)|=m,\; D\text{ is }T\text{-free} \right\}. \] The proof attaches two auxiliary digraphs to each palette and converts the palette optimization into a degree-square Tur\'an problem. We also prove a general majorization principle for convex out-degree moments in $F$-free digraphs. Whenever an ordinary Tur\'an extremal construction has extremal initial segments, the same construction maximizes every nondecreasing convex function of the out-degree sequence. Applying this to the Brown--Harary and Zhou--Li extremal digraphs for directed cycles gives exact formulas for all convex out-degree moments in $\overrightarrow{C}_{\ell}$-free digraphs. In particular, $\operatorname{ex}_2^+(m,\overrightarrow{C}_{3}) =\frac{m(m^2-1)}{3}.$ Consequently, for $m$ color the sharp density avoiding the two cyclic-triangle palettes is $\frac13-\frac1{3m^2}.$ Combining this exact finite endpoint with the palette classification theorem, we obtain finite $3$-graphs $H_m$ satisfying \[ \frac13-\frac1{3m^2} \le \pi_{\mathrm u}(H_m) \le \frac13. \] Thus the densities of these finite hypergraphs converge to $\frac13$.