Question 89
A set $S = \{s_1, \dots, s_k\}$ of positive real numbers is *polygonal* if $k \ge 3$ and there is a non-degenerate planar $k$-gon whose side lengths are exactly $s_1, \dots, s_k$; the set $S$ is *multipolygonal* if in every partition of $S$ into two subsets, each of which has at least three elements, exactly one of these two subsets is polygonal. Fix an integer $n \ge 7$.
a) Does there exist an $n$-element multipolygonal set, removal of whose maximal element leaves a multipolygonal set?
a) Does there exist an $n$-element multipolygonal set, removal of whose maximal element leaves a multipolygonal set?