The curse of the Euclidean metric: http://corner.mimuw.edu.pl/?p=1073
Krzysztof Fleszar posts about a big difficulty with algorithms for problems like Euclidean shortest paths where the answer is a sum of distances: we don't know how to compare two solutions efficiently.
Krzysztof's SODA 2019 paper on approximate TSP of hyperplanes (find a short tour that touches each given hyperplane) is https://epubs.siam.org/doi/10.1137/1.9781611975482.67 — fortunately in this case it's possible to use approximate numerical comparisons.
Paul Erdős died in 1996, but his most recent paper is from 2015, nearly 20 years later! There's a writeup at https://www.simonsfoundation.org/2015/12/10/new-erdos-paper-solves-egyptian-fraction-problem/; the paper itself is http://www.math.ucsd.edu/~ronspubs/pre_tres_egyptian.pdf
It's about Egyptian fractions – representations of rationals as sums of distinct unit fractions – and is motivated by the conjecture that it's always possible for all denominators to be semiprime. That's still open, but they prove that every integer has a representation with all denominators products of three primes.
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