AI 中文总结
本文解决了多部团问题中剩余的$K_4$-自由情形，确定了函数$f(n,7,4)$的精确值，并通过三色性结构论证得出其值约为$\lfloor30n/7\rfloor$。
AI 中文摘要
对于满足$2\le t\le r-1$的整数$n,r,t$，设$f(n,r,t+1)$表示大小为$n$的平衡$r$部图（不含$K_{t+1}$副本）的最小度的最大可能值。Lo、Treglown和Zhao在讨论$K_4$-自由族时，将$f(n,7,4)$确定为唯一剩余情形。我确定了该函数对每个$n\ge1$的值。首先，相应的三色极值$\delta(n,7,3)$被简化为一个$7\times3$整数矩阵问题并精确确定。其次，一个结构论证表明，每个平衡的$7$部$K_4$-自由图$G$，若满足$\delta(G)>\frac{132}{31}n$，则是三色的。因此，$f(n,7,4)=\lfloor30n/7\rfloor$，除非当$n\equiv4\pmod7$且$n\ge11$时，其值少1。
英文摘要
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.
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