Inverting the Turán Problem
Year: 2017
Type: preprint
Abstract: Classical questions in extremal graph theory concern the asymptotics of $\operatorname{ex}(G, \mathcal{H})$ where $\mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard' increasing sequence of host graphs $(G_1, G_2, \dots)$, most often $K_n$ or $K_{n,n}$. Inverting the question, we can instead ask how large $e(G)$ can be with respect to $\operatorname{ex}(G,\mathcal{H})$. We show that the standard sequences indeed maximize $e(G)$ for some choices of $\mathcal{H}$, but not for others. Many interesting questions and previous results arise very naturally in this context, which also, unusually, gives rise to sensible extremal questions concerning multigraphs and non-uniform hypergraphs.
Source: arXiv (Cornell University)
Authors: Briggs, Joseph, Cox, Christopher
Institution: Carnegie Mellon University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2963476322