On the minimum number of triangles in balanced tripartite graphs with large minimum degree

Abstract

Let $f(n,t)$ be the minimum number of triangles in a tripartite graph with $n$ vertices in each part and minimum degree at least $n+t$. In 1975, Bollob\'{a}s, Erd\H{o}s and Szemer\'{e}di proved that $f(n,1)=\min\{4,n\}$. They further remarked that it is ``very likely''that $f(n,t)\ge4t^3$ for $n\ge5t$. They also proved that $f(n,t) \ge t^3$ for all integers $n \ge t \ge 1$. We construct graphs showing that, for all integers $t\ge1$ and $n\ge3t+2\lceil(1+\sqrt5)t/2\rceil$, \[ f(n,t)\le(1+\sqrt5)t^3+\left(1+\frac1{\sqrt5}\right)t^2. \] Here $1+\sqrt5\approx3.236<4$, and the displayed upper bound is strictly less than $4t^3$ for every $t\ge2$, disproving their proposed bound. We also improve their lower bound $t^3$ by showing that $f(n,t)\ge\frac{12}{5}t^3$ for all integers $t\ge2$ and $n\ge18t^6$.