The Remaining $K_4$-Free Case in the Multipartite Clique Problem

Abstract

For integers $n,r,t$ with $2\le t\le r-1$, let $f(n,r,t+1)$ denote the largest possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and containing no copy of $K_{t+1}$. Lo, Treglown and Zhao identified $f(n,7,4)$ as the only remaining case in their treatment of the $K_4$-free family. I determine this function for every $n\ge1$. First, the corresponding three-colourable extremum $δ(n,7,3)$ is reduced to a $7\times3$ integer matrix problem and determined exactly. Second, a structural argument shows that every balanced $7$-partite $K_4$-free graph $G$ with $δ(G)>\frac{132}{31}n$ is three-colourable. Consequently, $f(n,7,4)=\lfloor30n/7\rfloor$ except when $n\equiv4\pmod7$ and $n\ge11$, where it is one less.