Constants

quantity	value	source
conjectured out-degree threshold	$1/3$	Caccetta–Häggkvist 1978
published unrestricted threshold (refereed)	$0.3465$	Hladký–Král'–Norin, Combinatorica 37 (2017)
claimed F₆ threshold	$0.3388$	de Joannis de Verclos–Sereni–Volec, Mar 2014, personal communication
two-sided threshold	$0.343545$	Lichiardopol 2010 (both $\delta^+$ and $\delta^-$ )
Shen elementary	$3-\sqrt{7}\approx 0.354249$	Shen, JCTB 74 (1998)
HHK	$0.3532$	Hamburger–Haxell–Kostochka, Electron. J. Combin. 2007
tonight, same F₄ system	$0.34645$	`certs/f4_certificate.json`, worst $F=-0.12343$
tonight, F₄ plus CKLS 2015 fork	$0.34640$	`q4/certs/keep/f4_or_new_certificate.json`, worst $F=-0.41903$
F₄ SDP saturation (this system)	$\approx 0.346439$	`certs/sdp_bound.json`

Cyclic construction: out-set $\{1,\ldots,\lfloor(n-1)/3\rfloor\}$ on $\mathbb{Z}/n$ , C₃-free.

Exact finite statement: every $n$ -vertex oriented graph with $\delta^+\ge\lceil n/3\rceil$ has a directed triangle.

First $n$ not implied by Hoàng–Reed ( $r\le 5$ ) plus HKN $0.3465$ was $n=18$ . That order, and the later holes $n=21,24,26,27,29,30,32,33,35,36$ , now have stored in-degree-cube DRATs.

Exact orders with stored in-degree-cube DRATs now include every leftover hole through $n=152$ .

First remaining hole: $n=153$ , $\delta^+=51$ .