Bulletin of the Korean Mathematical Society (대한수학회보)

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THE FRACTIONAL WEAK DISCREPANCY OF (M, 2)-FREE POSETS

  • Choi, Jeong-Ok (Division of Liberal Arts and Sciences Gwangju Institute of Science and Technology)
  • Received : 2017.07.28
  • Accepted : 2018.10.11
  • Published : 2019.01.31
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Abstract

For a finite poset $P=(X,{\preceq})$ the fractional weak discrepancy of P, denoted $wd_F(P)$, is the minimum value t for which there is a function $f:X{\rightarrow}{\mathbb{R}}$ satisfying (1) $f(x)+1{\leq}f(y)$ whenever $x{\prec}y$ and (2) ${\mid}f(x)-f(y){\mid}{\leq}t$ whenever $x{\parallel}y$. In this paper, we determine the range of the fractional weak discrepancy of (M, 2)-free posets for $M{\geq}5$, which is a problem asked in [9]. More precisely, we showed that (1) the range of the fractional weak discrepancy of (M, 2)-free interval orders is $W=\{{\frac{r}{r+1}}:r{\in}{\mathbb{N}}{\cup}\{0\}\}{\cup}\{t{\in}{\mathbb{Q}}:1{\leq}t<M-3\}$ and (2) the range of the fractional weak discrepancy of (M, 2)-free non-interval orders is $\{t{\in}{\mathbb{Q}}:1{\leq}t<M-3\}$. The result is a generalization of a well-known result for semiorders and the main result for split semiorders of [9] since the family of semiorders is the family of (4, 2)-free posets.

Keywords

E1BMAX_2019_v56n1_1_f0001.png 이미지

FIGURE 1. A (5, 2)-free poset P with wdF (P ) =$\frac{8}{5}$. The ele-ments of P form a forcing cycle with r = 8 and q = 5. Thevalues of f is written in the parenthesis. P contains 2 + 2 asa subposet but P has no induced 4 + 1 or 3 + 2.

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