The rainbow Erd\H{o}s-Rothschild problem for the Fano plane

TL;DR

This paper investigates the maximum number of r-edge colorings of large hypergraphs avoiding rainbow Fano plane copies, extending classical extremal problems in hypergraph coloring.

Contribution

It establishes that for large n and r, the balanced bipartite hypergraph maximizes rainbow Fano plane-free colorings, generalizing previous extremal results.

Findings

B_n maximizes the number of rainbow Fano plane-free colorings for large n and r.

The result extends classical extremal hypergraph theory to rainbow coloring contexts.

Identifies the structure of hypergraphs that optimize rainbow coloring constraints.

Abstract

The Fano plane is the unique linear 3-uniform hypergraph on seven vertices and seven hyperedges. It was recently proved that, for all $n \geq 8$, the balanced complete bipartite 3-uniform hypergraph on $n$ vertices, denoted by $B_n$, is the 3-uniform hypergraph on $n$ vertices with the largest number of hyperedges that does not contain a copy of the Fano plane. For sufficiently large $r$ and $n$, we show that $B_n$ admits the largest number of $r$-edge colorings with no rainbow copy of the Fano plane.