Generalizations of the Szemer\'edi-Trotter Theorem

TL;DR

This paper extends the Szemerédi-Trotter theorem to higher dimensions, providing tight bounds on the number of flags and exploring various incidence problems involving points, lines, and planes in 3D.

Contribution

It introduces a generalized upper bound for flags in higher dimensions and investigates new incidence problems in three-dimensional space, including group-theoretic interpretations.

Findings

Derived tight bounds for flags in higher dimensions.

Analyzed incidence problems with constraints in D space.

Connected incidence problems to group theory for new insights.

Abstract

We generalize the Szemer\'edi-Trotter incidence theorem, to bound the number of complete \emph{flags} in higher dimensions. Specifically, for each $i=0,1,\ldots,d-1$, we are given a finite set $S_i$ of $i$-flats in $\R^d$ or in $\C^d$, and a (complete) flag is a tuple $(f_0,f_1,\ldots,f_{d-1})$, where $f_i\in S_i$ for each $i$ and $f_i\subset f_{i+1}$ for each $i=0,1,\ldots,d-2$. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in $\R^3$ such that among the lines incident to a point, at most $O(1)$ of them can be coplanar, (ii) incidences with Legendrian lines in $\R^3$, a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in $\R^3$ (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems.