Tur\'an, involution and shifting

TL;DR

This paper strengthens Turán's (3,4)-conjecture using algebraic shifting, confirms its graph analogue, and generalizes the Mantel-Turán theorem by weakening its assumptions involving involutions.

Contribution

It introduces a new algebraic shifting approach to Turán's conjecture and extends the Mantel-Turán theorem to graphs with involutions under weaker conditions.

Findings

Algebraic shifting strengthens Turán's (3,4)-conjecture.

Graph analogue of the conjecture holds true.

Generalized Mantel-Turán theorem with involutions.

Abstract

We propose a strengthening of the conclusion in Tur\'an's (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel-Tur\'an theorem by weakening its assumption: for any graph G on n vertices and any involution on its vertex set, if for any 3-set S of the vertices, the number of edges in G spanned by S, plus the number of edges in G spanned by the image of S under the involution, is at least 2, then the number of edges in G is at least the Mantel-Tur\'an bound, namely the number achieved by two disjoint cliques of sizes n/2 rounded up and down.