The fractional weak discrepancy of $(M, 2)$-free posets

For a finite poset $P = (X, \preceq)$ the {\it fractional weak discrepancy} of $P$, denoted $wd_F(P)$, is the minimum value $t$ for which there is a function $f: X \longrightarrow \mathbb{R}$ satisfying (1) $f(x) + 1 \le f(y)$ whenever $x \prec y$ and (2) $|f(x) - f(y)| \le t$ whenever $x \| y$. In this paper, we determine the range of the fractional weak discrepancy of $(M, 2)$-free posets for $M \ge 5$, which is a problem asked in \cite{sst3}. More precisely, we showed that (1) the range of the fractional weak discrepancy of $(M, 2)$-free interval orders is $W = \{ \frac{r}{r+1} \colon r \in \mathbb{N} \cup \{ 0 \} \} \cup \{ t \in \mathbb{Q} \colon 1 \le t < M - 3 \}$ and (2) the range of the fractional weak discrepancy of $(M, 2)$-free non-interval orders is $\{ t \in \mathbb{Q} \colon 1 \le t < M - 3 \}$. The result is a generalization of a well-known result for semiorders and the main result for split semiorders of \cite{sst3} since the family of semiorders is the family of $(4, 2)$-free posets.