Turán, involution and shifting
Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 367-378.

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 n 2 rounded up and down.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.30
Keywords: Turán’s (3,4)-conjecture, shifting, threshold graphs
Kalai, Gil 1; Nevo, Eran 1

1 Einstein Institute of Mathematics The Hebrew University of Jerusalem Jerusalem (Israel)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Kalai, Gil; Nevo, Eran. Turán, involution and shifting. Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 367-378. doi : 10.5802/alco.30. https://alco.centre-mersenne.org/articles/10.5802/alco.30/

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