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Triforce and corners

Part of: Sequences and sets Extremal combinatorics

Published online by Cambridge University Press:  12 July 2019

JACOB FOX ,
ASHWIN SAH ,
MEHTAAB SAWHNEY ,
DAVID STONER  and
YUFEI ZHAO
JACOB FOX
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. e-mail: jacobfox@stanford.edu
ASHWIN SAH
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e-mail: asah@mit.edu, msawhney@mit.edu
MEHTAAB SAWHNEY
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e-mail: asah@mit.edu, msawhney@mit.edu
DAVID STONER
Affiliation:
Harvard University, Cambridge, MA 02138, U.S.A. e-mail: dstoner@college.harvard.edu
YUFEI ZHAO
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e-mail: yufeiz@mit.edu
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Abstract

May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).

Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).

On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Secondary: 05D05: Extremal set theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 1 , July 2020 , pp. 209 - 223
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

Supported by a Packard Fellowship and NSF Career Award DMS-1352121

‡

Supported by NSF Award DMS-1764176, and the MIT Solomon Buchsbaum Fund

References

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