Turán, involution and shifting

Gil Kalai,Eran Nevo

doi:10.5802/alco.30

Abstract

Turán, involution and shifting

Highlights

Let T (n) be a graph on the vertex set [n] = {1, 2, . . . , n} which is the disjoint union of two cliques of sizes n 2 and n 2Our starting point is the following influentialMantel–Turán theorem [18, 24].Theorem 1.1 (Mantel, Turán)
We relate B(n) to the Mantel–Turán theorem in two ways: first, by weakening the condition in Theorem 1.1 so that its conclusion applies to B(n) as well; second, by strengthening the conclusion in Theorem 1.1 to an algebraic one, namely that any graph G as in the theorem dominates B(n)
We prove this statement via exterior algebraic shifting, introduced by Kalai [11], and via relations between algebraic and combinatorial shifting established by Hibi and Murai [19, 20]. (We can apply algebraic shifting w.r.t. any fixed term order

Summary

Introduction

Let T (n) be a graph on the vertex set [n] = {1, 2, . Mantel–Turán theorem [18, 24]. Let G be a graph on [n] where every 3-subset of [n] contains an edge of G. (By a slight abuse of notation we refer to B(n) as the graph with the vertex set [n] and edge set B(n).) not every 3-subset of [n] contains an edge of B(n). We relate B(n) to the Mantel–Turán theorem in two ways: first, by weakening the condition in Theorem 1.1 so that its conclusion applies to B(n) as well; second, by strengthening the conclusion in Theorem 1.1 to an algebraic one, namely that any graph G as in the theorem dominates B(n).

Dominance and shifting

Turán with shifting

Turán with involution