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Two-colored transitive digraph has independent cover

Source: Kürschák 2016, problem 2

October 7, 2016

combinatorics

Problem Statement

Prove that for any finite set

A

of positive integers, there exists a subset

B

of

A

satisfying the following conditions: [*]if

b_1,b_2\in B

are distinct, then neither

b_1

and

b_2

nor

b_1+1

and

b_2+1

are multiples of each other, and [*] for any

a\in A

, we can find a

b\in B

such that

a

divides

b

or

b+1

divides

a+1

.

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