Suppose we have $n$ points $P$ and $m$ circles $C$ in the plane. Let $I(P,C)=\{(p,c), p \in P, c \in C, p \in c\}.$ Then what do we know about $\max_{m,n} |I(P,C)|$?

Any references?


I believe these are the best known upper and lower bounds, for $m$ points and $n$ circles [reversing the OP's notation to follow published convention], established in the two cited papers: $$ I(P,C) = O^*( m^\frac{2}{3} n^\frac{2}{3} + m^\frac{6}{11} n^\frac{9}{11}) + m + n) $$ $$ I(P,C) = \Omega^*( m^\frac{2}{3} n^\frac{2}{3} + m + n) $$ (where the $*$ hides logarithmic factors).

Marcus, Adam, and Gábor Tardos. "Intersection reverse sequences and geometric applications." In International Symposium on Graph Drawing, pp. 349-359. Springer, Berlin, Heidelberg, 2004.

Pach, János, and Micha Sharir. "Geometric incidences." Contemporary Mathematics 342 (2004): 185-224.

          Figure from Pach, Sharir, "Geometric incidences."

Complexity results have been extended to $\mathbb{R}^3$:

Sharir, Micha, Adam Sheffer, and Joshua Zahl. "Improved bounds for incidences between points and circles." Combinatorics, Probability and Computing 24, no. 3 (2015): 490-520. Preliminary arXiv version.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.