Geodesic
Vector sets with no repeated differences
Colloquium Mathematicum, Tome 64 (1993) no. 1, pp. 129-134

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DOI

We consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is "yes" if the set is of size at most $ℵ_2$, "not" if the set is allowed to be of size $(2^{2^{ℵ_0}})^{+}$. It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum. 
Péter Komjáth. Vector sets with no repeated differences. Colloquium Mathematicum, Tome 64 (1993) no. 1, pp. 129-134. doi: 10.4064/cm-64-1-129-134
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