# Incidences between points and circles in the plane

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.