On some non-rigid unit distance patterns (Q6624178)

Fix a natural number \(k\). The paper determines the maximum number of \(k\)-paths in unit distance graphs with \(n\) vertices on the sphere of radius \(1/\sqrt{2}\) within a polylogarithmic factor; and gives bounds for the maximum number of \(k\)-cycles in unit distance graphs with \(n\) vertices on the sphere of radius \(1/\sqrt{2}\), which are similarly tight for \(k\geq 10\). They also show that for a given \(3\)-regular graph on \(k\) vertices, the number of copies of this graph in unit distance graphs with \(n\) vertices in \(\mathbb{R}^3\) is at most a polylog factor times \(n^{k/2}\). All these results are closely related to incidence problems between points and lines.