0$. \[prop4\]

**Proof.** Let ${{\cal G}}_n$ denote the family of all graphs of order $n$. By Ramsey’s theory, for all graphs $G \in {{\cal G}}_n$ we have $\max \{ \alpha(G), \alpha({\overline{G}}) \} \ge c \log n$ for some constant $c$. For $r \ge 1$, let $G$ be an $r$-regular graph in ${{\cal G}}_n$. Then, $G$ contains either an independent set or a clique of order at least $c\log n$ for some constant $c$. Let $X$ be the vertex set of such an independent set or clique in $G$. Then, $|X| \ge c\log n$ and the subgraph $G[X]$ induced by $X$ is $s$-regular for some $s$. If $X$ is an independent set in $G$, then $s = 0$, while if $X$ is a clique, then $s = |X| - 1 \le r$. Further if $|X| - 1 = r$, then we note that $K_{r+1}$ is a component of $G$. On the one hand, if $s <
r$, then for every vertex $v \in X$, we have $|N(v) \cap (V
\setminus X)| = r-s > 0$ and hence $V \setminus X$ is a FD-set of $G$, and so ${{\rm fd}}(G) \le n - |X| \le n - c\log n$. On the other 