An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

Let P be a finite point set in the plane. A c-ordinary triangle in P is a subset of P consisting of three non-collinear points such that each of the three lines determined by the three points contains at most c points of P . Motivated by a question of Erdős, and answering a question of de Zeeuw, we prove that there exists a constant c > 0 such that P contains a c-ordinary triangle, provided that P is not contained in the union of two lines. Furthermore, the number of c-ordinary triangles in P is Ω(|P |).2010 Mathematics Subject Classification. 52C30.

Let F p be a prime field of order p > 2, and A be a set in F p with very small size in terms of p. In this note, we show that the number of distinct cubic distances determined by points in A × A satisfieswhich improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove thatthat is not of the form g(αx + βy) for some univariate polynomial g.

We prove bounds on intersections of algebraic varieties in C 4 with Cartesian products of finite sets from C 2 , and we point out connections with several classic theorems from combinatorial geometry. Consider an algebraic variety X ⊂ C 4 of degree d, such that not all polynomials that vanish on X are of the formwhere G, H, K, L are polynomials and G and K are not constant. Let P, Q ⊂ C 2 be finite sets of size n. If X has dimension one or two, then we prove |X ∩ (P × Q)| = O d (n), while if X has dimension three, then |X ∩ (P × Q)| = O d,ε (n 4/3+ε ) for any ε > 0. Both bounds are best possible in this generality (except for the ε).These bounds can be viewed as different generalizations of the Schwartz-Zippel lemma, where we replace a product of "one-dimensional" finite subsets of C by a product of "twodimensional" finite subsets of C 2 . The bound for three-dimensional varieties generalizes the Szemerédi-Trotter theorem. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz.As corollaries of our two bounds, we obtain bounds on the number of repeated and distinct values of polynomials and polynomial maps of pairs of points in C 2 , with a characterization of those maps for which no good bounds hold. These results generalize known bounds on repeated and distinct Euclidean distances.

According to Suk's breakthrough result on the Erdős-Szekeres problem, any point set in general position in the plane, which has no n elements that form the vertex set of a convex n-gon, has at most 2 n+O(n 2/3 log n) points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of n convex bodies in the plane is said to be in convex position if the convex hull of the union of no n − 1 of its members contains the remaining one. If any three members are in convex position, we say that the family is in general position. Combining our results with a theorem of Dobbins, Holmsen, and Hubard, we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes Tóth and by Pach and Tóth, respectively. Let c(n) (and c (n)) denote the smallest positive integer N with the property that any family of N pairwise disjoint convex bodies in general position (resp., N convex bodies in general position, any pair of which share at most two boundary points) has an n-membered subfamily in convex position. We show that c(n) ≤ c (n) ≤ 2 n+O(√ n log n) .