MathDB

Let

p_1,\dots,p_k

be prime numbers, and let

S

be the set of those integers whose all prime divisors are among

p_1,\dots,p_k

. For a finite subset

A

of the integers let us denote by

\mathcal G(A)

the graph whose vertices are the elements of

A

, and the edges are those pairs

a,b\in A

for which a \minus{} b\in S. Does there exist for all

m\geq 3

m

-element subset

A

of the integers such that (i)

\mathcal G(A)

is complete? (ii)

\mathcal G(A)

is connected, but all vertices have degree at most 2?

2