Paradoxes of two-length interval orders (Q2266725)

The concept of a two-length interval order is introduced. With the shorter length fixed at 1, let L denote the set of admissible longer lengths for which this concept holds for a given interval order. The paper demonstrates that there are two-length interval orders on finite point sets with the following sets L for each integer \(m\geq 2:\) \(L=(1,m)\); \(L=(2-1/m,2)\cup (m,\infty)\); \(L=(m,2m-1)\cup (2m-1,\infty)\). The second case shows that L can have an arbitrarily big gap between admissible longer length, and the third case leads to the corollary that there can be arbitrarily many gaps or holes in L.