Rainbow saturation and graph capacities

TL;DR

This paper proves a conjecture about the asymptotic behavior of rainbow saturation numbers for triangles, showing they grow proportionally to n log n, using probabilistic and information theory methods.

Contribution

It confirms the conjecture that rainbow saturation numbers for triangles are asymptotically proportional to n log n, providing tight bounds and novel proof techniques.

Findings

Rainbow saturation number for triangles is Θ(n log n).

Lower bound derived using probabilistic methods.

Upper bound established via Shannon capacity of clique families.

Abstract

The $t$-colored rainbow saturation number $rsat_t(n,F)$ is the minimum size of a $t$-edge-colored graph on $n$ vertices that contains no rainbow copy of $F$, but the addition of any missing edge in any color creates such a rainbow copy. Barrus, Ferrara, Vandenbussche and Wenger conjectured that $rsat_t(n,K_s) = \Theta(n\log n)$ for every $s\ge 3$ and $t\ge \binom{s}{2}$. In this short note we prove the conjecture in a strong sense, asymptotically determining the rainbow saturation number for triangles. Our lower bound is probabilistic in spirit, the upper bound is based on the Shannon capacity of a certain family of cliques.