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Statement

Setting. $\delta(G,\tau)$ is the co-degree function of Definition 3.2 (p. 11), recalled on the page of Theorem 3.4; $e(G[C])$ is the number of edges of $G$ inside $C$.

Corollary 3.6 (p. 14). Let $G$ be an $r$-graph on vertex set $[n]$, let $0<\epsilon,\tau<1/2$, and suppose that $\delta(G,\tau)\le\epsilon/12r!$. Then there are a constant $c=c(r)$ and a function $C:\mathcal P([n])^s\to\mathcal P[n]$, with $s\le c\log(1/\epsilon)$, such that, writing $\mathcal T={(T_1,\ldots,T_s)\in\mathcal P([n])^s:|T_i|\le c\tau n,
1\le i\le s}$ and $\mathcal C=\{C(T):T\in\mathcal T\}$,

• (a) every independent set $I$ has some $T=(T_1,\ldots,T_s)\in\mathcal T\cap\mathcal P(I)^s$ with $I\subset C(T)\in\mathcal C$;
• (b) $e(G[C])\le\epsilon e(G)$ for every $C\in\mathcal C$;
• (c) $\log|\mathcal C|\le c\log(1/\epsilon)n\tau\log(1/\tau)$.

Property (a) holds as well for every $I\subset[n]$ such that $G[I]$ is $\lfloor\epsilon\tau^{r-1}e(G)/12r!n\rfloor$-degenerate or $e(G[I])\le24\epsilon r!r\tau^re(G)$.

Remark (p. 15). Where the constant matters, the paper says that $c(r)=800r!^3r$ can be taken.

Source. David Saxton and Andrew Thomason, Hypergraph containers, Invent. Math. 201 (2015), 925--992; arXiv:1204.6595. Labels and pages here are those of arXiv:1204.6595v3: the corollary on p. 14, its proof on p. 32 (Section 6, pp. 29--32). The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof was read but not checked step by step.

Proof pointer

Section 6. Theorem 6.2 (p. 30) applies Theorem 3.4 with $\zeta=1/12r!$ to get containers with $e(G[C])\le(1-1/2r!)e(G)$, counted by Lemma 6.1 (p. 30). Theorem 6.3 (p. 31) iterates Theorem 6.2 inside each container until it spans at most $e_0$ edges. The corollary (proof on p. 32) is Theorem 6.3 with $e_0=\epsilon e(G)$ and $\tau(U)=\tau$ for every $U$, which is allowed since $e(G[U])\ge\epsilon e(G)$ gives $\delta(G[U],\tau)\le\delta(G,\tau)/\epsilon \le1/12r!$; the count is

$$\log|\mathcal C|\le288r!^2r\Bigl(1+\frac{\log\epsilon}{\log(1-1/2r!)}\Bigr)n\tau\log(1/\tau).$$

Dependencies

Theorem 3.4 (p. 13); Lemma 6.1, Theorem 6.2 and Theorem 6.3 (pp. 30--31).

Bears on

No Erdős problem is linked from this result. The paper uses it to prove Theorem 2.3 (through Theorem 9.2) and the regular case of Theorem 2.1.