Constructions of point-line arrangements in the plane with large girth

TL;DR

This paper explores geometric point-line arrangements with large girth, providing new lower bounds on incidences by adapting finite geometry constructions, and investigates the tightness of extremal cycle bounds in incidence graphs.

Contribution

It introduces novel geometric constructions that establish lower bounds on incidences in arrangements with large girth, advancing understanding of extremal properties in geometric graph theory.

Findings

Constructed arrangements with girth at least k+5 and Ω(n^{1+4/(k^2+6k-3)}) incidences.

Modified known finite geometry constructions to improve lower bounds.

Applied techniques to Wenger graphs for k=5, achieving better bounds.

Abstract

A classical result by Erd\H{o}s, and later on by Bondy and Simonivits, states that every $n$-vertex graph with no cycle of length $2k$ has at most $O(n^{1+1 /k})$ edges. This bound is known to be tight when $k \in \{2,3,5\},$ but it is a major open problem in extremal graph theory to decide if this bound is tight for all $k$. In this paper, we study the effect of forbidding short even cycles in incidence graphs of point-line arrangements in the plane. It is not known if the Erd\H{o}s upper bound stated above can be improved to $o(n^{1+1/k})$ in this geometric setting, and in this note, we establish non-trivial lower bounds for this problem by modifying known constructions arising in finite geometries. In particular, by modifying a construction due to Labeznik and Ustimenko, we construct an arrangement of $n$ points and $n$ lines in the plane, such that their incidence graph has girth at least $k + 5$, and determines at least $\Omega({n^{1+\frac{4}{k^2+6k-3}}})$ incidences. We also apply the same technique to Wenger graphs, which gives a better lower bound for $k=5.$