Judicious partitions for restricted self-sumsets in cyclic groups

We study the minimax problem for restricted two-fold self-sumsets in $k$-colorings of ${\mathbb{Z}}_n$. For primes $p$ with $2\le k\le p$ we determine the exact minimum $\max \{0,2\lceil p\mathrm{/}k\rceil -3\}$. For general $n$ (with $m=\lceil n\mathrm{/}k\rceil$) we bound the optimum between a size term $\min \{p(n),2m-3\}$ and a periodicity term $f(n\mathrm{/}q(n,k))$, and show these bounds are tight when $2m-3\le p(n)$ or $f(n\mathrm{/}q(n,k))\le \min \{p(n),2m-3\}$. We further prove a stability inequality and a threshold theorem that force concentration in a single subgroup coset near the periodic scale. In the prime case with $m\ge 5$ and $2m-3<p$, every optimal coloring contains a class of size $m$ that is an arc (an arithmetic progression up to an affine automorphism). Our approach combines the restricted Erdős–Heilbronn phenomenon with block/coset colorings and an injectivity window.