An improved constant factor for the unit distance problem
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We prove that the number of unit distances among n planar points is at most 1.94· n^4/3, improving on the previous best bound of 8n^4/3. We also give better upper and lower bounds for several small values of n. Our main method is a crossing lemma for multigraphs with a better constant, which is of independent interest, as our proof is simpler than earlier proofs.
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