Improved Bounds for Pencils of Lines
Oliver Roche-Newton, Audie Warren

TL;DR
This paper improves the upper bounds on the number of four-line intersection points in four pencils of lines in the plane and provides new constructions that surpass previous lower bounds, advancing combinatorial geometry understanding.
Contribution
It establishes a tighter upper bound of O(n^{11/6}) for four-pencil intersection points and introduces constructions that improve lower bounds for rich point counts.
Findings
Maximum four-pencil intersection points is O(n^{11/6})
Constructed arrangements with more four-rich points than previous lower bounds
Provided general position constructions with Omega_m(n^{3/2}) m-rich points
Abstract
We consider a question raised by Rudnev: given four pencils of concurrent lines in , with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is , improving a result of Chang and Solymosi. We also consider constructions for this problem. Alon, Ruzsa and Solymosi constructed an arrangement of four non-collinear -pencils which determine four-rich points. We give a construction to show that this is not tight, improving this lower bound by a logarithmic factor. We also give a construction of a set of -pencils, whose centres are in general position, that determine -rich points.
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