Claims

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1990_09_01_erdos_pach: Among n pairwise disjoint translates of one compact convex planar set, at most O(n^(4/3)) pairs are at distance exactly one.

2006_03_31_valtr: An Oberwolfach abstract of 2005 announces h(n) = Theta(n^(4/3)) for disjoint translates of a compact convex set, also with centrally symmetric translates, outlining the construction only in the talk.

2026_04_27_chojecki: A small parabolic-cap body gives n disjoint translates with at least c n^(4/3) unit distances, so with Erdős and Pach h(n) = Theta(n^(4/3)) and any c < 1/3 answers the question.

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