TY - JOUR

T1 - Turán, involution and shifting

AU - Kalai, Gil

AU - Nevo, Eran

N1 - Funding Information:
Acknowledgements. Research of Kalai is partially supported by ERC advanced grant 320924, BSF grant 2006066, and NSF grant DMS-1300120, and of Nevo by Israel Science Foundation grant ISF-1695/15, by grant 2528/16 of the ISF-NRF Singapore joint research program, and by ISF-BSF joint grant 2016288.
Publisher Copyright:
© The journal and the authors, 2019.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We propose a strengthening of the conclusion in Turán’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–Turán 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–Turán bound, namely the number achieved by two disjoint cliques of sizes n2 rounded up and down.

AB - We propose a strengthening of the conclusion in Turán’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–Turán 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–Turán bound, namely the number achieved by two disjoint cliques of sizes n2 rounded up and down.

KW - Shifting

KW - Threshold graphs

KW - Turán’s (3,4)-conjecture

UR - http://www.scopus.com/inward/record.url?scp=85111608738&partnerID=8YFLogxK

U2 - 10.5802/ALCO.30

DO - 10.5802/ALCO.30

M3 - Article

AN - SCOPUS:85111608738

SN - 2589-5486

VL - 2

SP - 367

EP - 378

JO - Algebraic Combinatorics

JF - Algebraic Combinatorics

IS - 3

ER -