It is consistent that there exists a graph X of cardinality $ℵ_1$ such that every graph has an edge coloring with $ℵ_1$ colors in which the induced copies of X (if there are any) are totally multicolored (get all possible colors).
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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.
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