A smallest-fibre-size to poset-size ratio approaching \({8 \over 15}\) (Q1200010)

Let \(P\) be a finite poset with no splitting elements and \(F\) be a fibre of \(P\) of minimal cardinality. There is the problem what is the greatest possible value of \(| F|/| P|\). \textit{D. Duffus}, \textit{H. A. Kierstead} and \textit{W. T. Trotter} [J. Comb. Theory, Ser. A 58, 158- 164 (1991; Zbl 0757.06001)] have shown that this value cannot exceed \({2\over 3}\). \textit{D. Duffus}, \textit{B. Sands}, \textit{N. Sauer} and \textit{R. Woodrow} [J. Comb. Theory, Ser. A 57, 109-116 (1991; Zbl 0742.06004)] have given a poset with \(| F|/| P|={9\over 17}\). In the paper under review, the author finds posets that show that a value of \((8n+1)/(15n+2)\) can be achieved for any positive integer \(n\).