A Note on a Problem of Erdős About Rich Distances

A question of Erdős asked whether there exists a set of 𝑛 points such that 𝑐 ⋅ 𝑛 distances occur more than 𝑛 times. We provide an affirmative answer to this question, showing that there exists a set of 𝑛 points such that distances occur more than 𝑛 times. We also present a generalized version, finding a set of 𝑛 points where 𝑐 𝑚 ⋅ 𝑛 distances occurring more than 𝑛 + 𝑚 times.