Tight paths in convex geometric hypergraphs

TL;DR

This paper proves a sharp theorem on tight paths in convex geometric hypergraphs, generalizing classical results and improving bounds on the Turán problem for such hypergraphs.

Contribution

It introduces a new geometric theorem that generalizes earlier results and provides the first significant improvement on the Turán problem for tight paths in uniform hypergraphs.

Findings

Theorem on tight paths in convex geometric hypergraphs is asymptotically sharp.

Generalizes classical results for convex geometric graphs.

Provides improved bounds for the Turán problem.

Abstract

In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz, Sutherland, Kupitz and Perles for convex geometric graphs, as well as the classical Erd\H{o}s-Gallai Theorem for graphs. As a consequence, we obtain the first substantial improvement on the Tur\'{a}n problem for tight paths in uniform hypergraphs.