MathDB

Let M \equal{} \lbrace 2, 3, 4, \ldots\, 1000 \rbrace. Find the smallest

n \in \mathbb{N}

such that any

n

-element subset of

M

contains 3 pairwise disjoint 4-element subsets

S, T, U

such that I. For any 2 elements in

S

, the larger number is a multiple of the smaller number. The same applies for

T

and

U

. II. For any

s \in S

and

t \in T

, (s,t) \equal{} 1. III. For any

s \in S

and

u \in U

(s,u) > 1

Problem Statement