K_4-free graphs have sparse halves

TL;DR

This paper proves that in any $K_4$-free graph with n vertices, there exists a large subset of about half the vertices with relatively few edges, specifically at most n^2/18 edges.

Contribution

It establishes a new upper bound on the number of edges in a large half of a $K_4$-free graph, advancing understanding of sparse structures within such graphs.

Findings

Existence of a half-vertex subset with at most n^2/18 edges

Improved bounds on edge density in $K_4$-free graphs

Structural insights into sparse subgraphs

Abstract

Every $K_4$-free graph on $n$ vertices has a set of $\lfloor n/2\rfloor$ vertices spanning at most $n^2/18$ edges.