Zarankiewicz问题的半代数形式
A semi-algebraic version of Zarankiewicz's problem
摘要：
一个二部图$G$在$\mathbb{R}^d$中是半代数的，如果它的顶点由点集$P,Q \subset \mathbb{R}^d$表示，它的边被定义为点对$(p,q) \in P\times Q$，它们满足$2d$坐标中固定数量的多项式方程和不等式的布尔组合。我们证明了对于固定的$k$，$|P| = m$和$|Q| = n$的$\mathbb{R}^2$中的无$K_{k,k}$的半代数二部图$G = (P,Q,E)$的最大边数至多为$O((mn)^{2/3} + m + n)$，并且这个界是紧的.在$d \geq 3$维中，我们证明了所有这样的半代数图最多有$C\left((mn)^{ \frac{d}{d+1} + \varepsilon} + m + n\right)$边，这里$\varepsilon$是一个任意小的常数，而$C = C(d,k,t,\varepsilon)$。这个结果是经典的Szemerédi-Trotter关联定理的一个意义深远的推广。证明结合了几个领域的工具：VC维和破碎函数，多项式分割和希尔伯特多项式。我们还提出了我们的定理的各种应用。例如，$\mathbb{R}^d$中的一般点簇关联界，Erdos单位距离问题的$d$维变体的改进界，等等。
英文摘要：
A bipartite graph $G$ is semi-algebraic in $\mathbb{R}^d$ if its vertices are represented by point sets $P,Q \subset \mathbb{R}^d$ and its edges are defined as pairs of points $(p,q) \in P\times Q$ that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in $2d$ coordinates. We show that for fixed $k$, the maximum number of edges in a $K_{k,k}$-free semi-algebraic bipartite graph $G = (P,Q,E)$ in $\mathbb{R}^2$ with $|P| = m$ and $|Q| = n$ is at most $O((mn)^{2/3} + m + n)$, and this bound is tight. In dimensions $d \geq 3$, we show that all such semi-algebraic graphs have at most $C\left((mn)^{ \frac{d}{d+1} + \varepsilon} + m + n\right)$ edges, where here $\varepsilon$ is an arbitrarily small constant and $C = C(d,k,t,\varepsilon)$. This result is a far-reaching generalization of the classical Szemerédi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials. We also present various applications of our theorem. For example, a general point-variety incidence bound in $\mathbb{R}^d$, an improved bound for a $d$-dimensional variant of the Erdős unit distances problem, and more.
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