An exact small-$n$ computation of the minimum 2-coloring discrepancy of $K_n^{(3)}$

TL;DR

This paper computes exact small-$n$ minimum 2-coloring discrepancy values for Steiner triple systems, providing new computational results, structural insights, and a conjectural formula, advancing understanding of discrepancy in combinatorial designs.

Contribution

It offers the first exact small-$n$ discrepancy values, structural analysis of low-discrepancy colorings, and a conjectural formula for $ ext{delta}_2(n)$ for specific $n$.

Findings

Exact $ ext{delta}_2(n)$ values for $n otin ext{small set}$ matching a derived formula.

Existence of a wide basin of near-optimal 2-colorings at $n=9$.

Random colorings have discrepancy growth consistent with a Gaussian heuristic.

Abstract

For an integer $r \ge 2$ and an order $n \equiv 1, 3 \pmod{6}$, write $\delta_r(n)$ for the minimum, over all $r$-colourings $\chi : \binom{[n]}{3} \to [r]$, of $\max_{\mathcal{S}} \mathrm{disc}(\mathcal{S}, \chi)$, where the maximum is over labelled Steiner triple systems $\mathcal{S}$ of order $n$ and $\mathrm{disc}(\mathcal{S}, \chi) = \max_c |\#\{T \in \mathcal{S} : \chi(T) = c\} - |\mathcal{S}|/r|$. Following Gishboliner, Glock, and Sgueglia \cite{GishbolinerGlockSgueglia2025}, the bulk of the recent work on this quantity has been on lower bounds for $r \ge 3$ (proving $\delta_r(n) = \Omega(n^2)$) and on structural characterisation of the low-discrepancy 2-colourings. We give three small computational contributions in the small-$n$ regime $n \in \{7, 9, 13, 15, 19, 21\}$: An exact value of $\delta_2(n)$ for each such $n$, matching the formula $\delta_2(n) = \min_{x \in [0, n] \cap \mathbb{Z}} |x(n-x)/2 - n(n-1)/12|$ obtained by optimising the GGS Example 1.1 family. Rigorous for $n \in \{7, 9\}$ via exhaustive search over labelled STSs ($30$ resp. $840$ systems) and over all $2$-colourings; computational for $n \in \{13, 15, 19, 21\}$ by simulated-annealing search; A wide near-optimal basin: at $n = 9$, every two-colour-flip neighbour of the optimal Example~1.1 colouring that maintains discrepancy $1.0$ exists; about $34\%$ of two-flip perturbations preserve optimality; Random-colouring statistics for $r \in \{2, 3, 4\}$: $\langle\max_{\mathcal{S}}\mathrm{disc}\rangle$ grows linearly in $n$, in agreement with a heuristic Gaussian estimate $n / \sqrt{6r} \cdot \sqrt{2 \log K}$ over $K$ sampled labellings; the typical-case discrepancy is far below the GGS worst-case $\Omega(n^2)$. We additionally state a conjectural exact formula for $\delta_2(n)$ that holds for every $n \equiv 1, 3 \pmod{6}$.