MathDB

(a) Prove that for every natural number

k

, there are positive integers

a_1<a_2<\ldots <a_k

such that

a_i-a_j

divides

a_i

for all

1\leq i, j\leq k, i\neq j

.
(b) Show that there is an absolute constant

C>0

such that

a_1>k^{Ck}

for every sequence

a_1,\ldots, a_k

of numbers that satisfy the above divisibility condition. [A. Balogh, I. Z. Ruzsa]

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