Erdösproblem 3, part II

I checked my dictionary, but apparently there is no english word for the dutch word ‘tangconstructie’. Anyway, remember how I, rhetorically, asked ‘how else could it be?’ yesterday? Well, I am more than delighted to say that I am able to give a counterexample to the metaconjecture that every proof of a counterexample to the question of Erdös and Graham I talked about in the previous post, works via a case-by-case-argument.

Consider the set X = \{2,3,4,5,6,8,9,10,12,15,16\} and let lcm\{S\} be the least common multiple of all members of S, for a set S of positive integers. Then \displaystyle \sum_{x_i \in X} \dfrac{1}{x_i} = 2 - \dfrac{1}{lcm\{X\}}. Now, assume we have a partition of X in two sets A and B with \displaystyle \sum_{a_i \in A} \dfrac{1}{a_i} < 1 and \displaystyle \sum_{b_i \in B} \dfrac{1}{b_i} < 1. Then in particular we have:

2 - \dfrac{1}{lcm\{X\}}
= \displaystyle \sum_{x_i \in X} \dfrac{1}{x_i}
= \displaystyle \sum_{a_i \in A} \dfrac{1}{a_i} + \displaystyle \sum_{b_i \in B} \dfrac{1}{b_i}
\le 1 - \dfrac{1}{lcm\{A\}} + 1 - \dfrac{1}{lcm\{B\}}
\le 2 - \dfrac{2}{lcm\{X\}}

And boom goes the dynamite.


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