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Statement

Conventions (p. 5): a tree $F$ viewed as a rooted tree has as its roots exactly its leaves. Definitions 12 and 13 (pp. 5--6): $\mathrm{Inj}(F,G)$ is the set of embeddings of $F$ into $G$ (injective maps of vertices sending edges to edges), and an embedding $\eta$ is $C$-ample if there are $C$ embeddings that agree with $\eta$ on $R(F)$ and map $V(F)\setminus R(F)$ to pairwise disjoint sets; $\mathrm{amp}_C(F,G)$ counts the $C$-ample embeddings. Definition 14 (p. 6): $F_2$ contains $F_1$ as a rooted subgraph if some embedding of $F_1$ into $F_2$ sends a vertex to a root of $F_2$ exactly when the vertex is a root of $F_1$.

Definition 15 (Obstruction family, p. 6, quoted). "Given a tree $F$, a family $\mathcal F_0$ of trees is an obstruction family for $F$ if every member of $\mathcal F_0$ is isomorphic to a subtree of $F$ that is not a single edge, and moreover for every nonempty proper subset $U$ of $V(F)\setminus R(F)$, after adding $U$ to the root set of $F$, the resulting rooted graph contains a member of $\mathcal F_0$ as a rooted subgraph."

Definition 16 (Negligible obstruction, p. 6, quoted). "Given two trees $F_0$ and $F$, we say that $F_0$ is negligible for $F$ if for every $p\in\mathbb N^+$ and $\varepsilon>0$ there exist $c_0>0$ and $C_0\in\mathbb N$ such that the following holds. For every $c>c_0$ and every $n$-vertex graph $G$ with $n\ge n_0(c)$, if every vertex in $G$ has degree between $d$ and $Kd$, where $d=cn^\alpha$, $K=5^{4/\alpha}$ and $\alpha=1-1/\rho_F$, and moreover $\mathrm{amp}_p(F,G)=0$, then $\mathrm{amp}_{C_0}(F_0,G)\le\varepsilon nd^{e(F_0)}$. An obstruction family for $F$ is negligible if every member of the family is negligible for $F$."

Lemma 17 (Negligibility lemma, p. 7, quoted). "Given a tree $F$, if there exists a negligible obstruction family $\mathcal F_0$ for $F$, then $\mathrm{ex}(n,F^p)=O(n^{2-1/\rho_F})$ for every $p\in\mathbb N^+$."

The lemma does not assume that $F$ is balanced. The paper proposes it as a two-step strategy for the Bukh--Conlon conjecture: find an obstruction family, then certify that it is negligible (pp. 4 and 7), and traces the ideas of the framework to Conlon and Lee (p. 3).

Source. T. Jiang, Z. Jiang and J. Ma, Negligible obstructions and Turán exponents, arXiv:2007.02975v3 (30 January 2023), Definitions 12--16 on pp. 5--6, Lemma 17 on p. 7, its proof in Section 3 (pp. 8--10); published in Ann. Appl. Math. 38 (2022), no. 3, 356--384, doi:10.4208/aam.OA-2022-0008, which was not compared. The edition read is identified in the source digest.

Read depth. Claims checked: Definitions 12--16 and Lemma 17 were read on the page images of pp. 5--7; the proof in Section 3 was read for structure only.

Proof pointer

Section 3 (pp. 8--10). By Lemma 20 (p. 8), a variant of the Erdős--Simonovits regularization that the paper takes from Bukh and Jiang, it suffices to find $F^p$ in an $n$-vertex graph whose degrees lie between $cn^\alpha$ and $Kcn^\alpha$. Assuming no $p$-ample embedding of $F$, the negligibility of each member of $\mathcal F_0$ removes few embeddings of $F$ that extend an ample embedding of an obstruction; a backward induction over subsets of $V(F)\setminus R(F)$, using Definition 15, bounds the remaining embeddings that fix the roots, and the pigeonhole principle then contradicts their number for $c$ large.

Dependencies

Lemma 20 (p. 8), which the paper attributes to Bukh and Jiang, its [4] (Theorem 12, arXiv version only).

Bears on

• Problem 571: through Theorem 8, whose proof applies the lemma to the trees $T_{s,t,s'}$.