MathDB

A set

S=\{ s_1,s_2,...,s_k\}

of positive real numbers is "polygonal" if

k\geq 3

and there is a non-degenerate planar

k-

gon whose side lengths are exactly

s_1,s_2,...,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 in polygonal. Fix an integer

n\geq 7

. (a) Does there exist an

n-

element multipolygonal set, removal of whose maximal element leaves a multipolygonal set? (b) Is it possible that every

(n-1)-

element subset of an

n-

element set of positive real numbers be multipolygonal?

Problem Statement