Open questions beyond the seven-cycle threshold

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These are the further questions of Section 8 of Plasma AI's write-up The seven-cycle case of the maximal anti-Ramsey conjecture, held as c7_threshold.pdf in the research evidence. None is needed for Problem 809, which the threshold theorem answers. Here $\chi_S(n,e,C_7)$ is the least number of colors in an edge coloring of some $n$-vertex graph with $e$ edges in which every seven-cycle is rainbow, and a family of edges is $C_7$-compatible when every two of its edges lie on a common seven-cycle.

1. The interior coefficient. For fixed $1/4<q<1/2$ and $s$ as in the five-clique construction, is $\chi_S(n,\lfloor qn^2\rfloor,C_7)=(q-s^3)n^2+o(n^2)$? The density lower bound is strictly smaller at every interior density, so the coefficient is open between the two.
2. Stability at the threshold. If $G_n$ has $\lfloor n^2/4\rfloor+1$ edges and a coloring with $n^2/8+o(n^2)$ colors in which every $C_7$ is rainbow, must $G_n$ be within $o(n^2)$ edge additions and deletions of two disjoint cliques of orders $n/2+o(n)$? The write-up gives a partial answer: the reweighting step of Section 4 of the solution note, run with cleaning parameter $\eta=v^2/128$ and weight shift $v/16$ when the normalized degree variance is at least $v>0$, raises the weighted density to at least $1/4+v^2/64$, and the finite inequality then forces more than $(1/8+v/70)n^2$ colors for large $n$. So the variance of such graphs tends to zero. That consequence is the write-up's and has not been checked here.
3. Compatible edge families. Does every $n$-vertex graph with more than $\lfloor n^2/4\rfloor$ edges contain a $C_7$-compatible family of at least $n^2/8-o(n^2)$ edges? A positive answer would strengthen the threshold theorem, since such a family needs that many colors. The near-regular and near-bipartite notes build such families under their hypotheses; the fractional coloring argument of palette savings does not produce one.
4. The error term. Is $\chi_S(n,\lfloor n^2/4\rfloor+1,C_7)=n^2/8+O(n)$? As the write-up notes, two cliques sharing one vertex already give an upper bound with this error term; the two disjoint cliques of Section 5 of the solution note give $n^2/8+O(n^{3/2})$.
5. A proof without regularity. Can the threshold theorem be proved without the Szemerédi regularity lemma, in particular by a direct structural argument in place of the passage from closed walks to cycles in seven-walk cleaning? Such an argument could improve the quantitative bounds. Bucić, Chen and Ma prove the longer-cycle cases by induction and short-path arguments (their Theorem 1.2).