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Statement

Section 5.1, p. 14, on partially defined 3-graphs represented as red-blue vertex-colored 3-graphs: "We have used such partially defined graphs to improve the bound of $\pi(K_4^3)$, the Turán density of the complete 3-graph on 4 vertices. The best known bound was held by Razborov [18] at 0.56167 by considering 3-graphs of order 6. We can decrease this to 0.5615 by looking at red-blue vertex-coloured 3-graphs of order 6, together with regularity constraints as described by Hladký, Král', and Norine [13]. We will describe the regularity constraints in more detail in Section 5.1.1. The relevant data required to prove the 0.5615 bound can be found in K4.txt located in the source files section on the arXiv, see [4]." Section 5.1.1 opens: "Our proof that $\pi(K_4^3)\le0.5615$ involves regularity constraints such as those described by Hladký, Král', and Norine for digraphs [13]." The paper adds that "a significantly better bound may be possible", since the colors of the non-labeled vertices were ignored for lack of time. The bound is stated in the text, not as a numbered theorem; the abstract announces it as "a new upper bound of 0.5615 for $\pi(K_4^3)$".

Source. R. Baber, Turán densities of hypercubes, arXiv:1201.3587v2 (13 November 2012, 18 pages; v1 17 January 2012; the arXiv record read says v2 was "Revised to include a new bound for $\pi(K_4^3)$" and carries no journal reference), the copy read, whose title page is dated 4 November 2018 by its typesetting; the passage is on p. 14 of Section 5.1 (pp. 13–16), read on the page image and in the text layer. A preprint; no refereed version was found. The edition read is identified in the source digest.

Read depth. Claims checked: the passage and the opening of Section 5.1.1 were read clause by clause on the page image. The certificate K4.txt was not fetched, the semidefinite argument was not checked, and the regularity-constraint argument of Section 5.1.1 was not read beyond its first paragraph.

Proof pointer

A flag-algebra (semidefinite) certificate over red-blue vertex-colored 3-graphs of order six with regularity constraints (Section 5.1.1); the numerical data are the file K4.txt in the arXiv source. Not checked here.

Dependencies

Razborov's flag algebra method and the regularity constraints of Hladký, Král' and Norine (external, at statement level); the certificate file.

Bears on

• Problem 500: the upper bound $\pi(K_4^3)\le0.5615$ on the Turán density the problem's asymptotic form asks for, above Turán's conjectured $5/9$, which it leaves open; a preprint result whose certificate was not checked here. It is below the $0.561666$ that Razborov's preprint records only as a numerical suggestion, a figure this paper writes as $0.56167$.