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Statement

Definitions (p. 2). A rooted graph is a graph $F$ with a set $R(F)$ of vertices called roots (Definition 4). Its density is $\rho_F=e(F)/(v(F)-|R(F)|)$, and $F$ is balanced if $\rho_F>1$ and, for every subset $S$ of $V(F)\setminus R(F)$, at least $\rho_F|S|$ edges of $F$ have an endpoint in $S$ (Definition 5).

The tree $T_{s,t,s'}$ (Figure 1, p. 3), described here in words: a centre joined to $t$ middle vertices, each of which has $s$ leaf neighbours, and to $s'$ further leaves; the roots are the $st+s'$ leaves. Its density is $\rho_F=(st+t+s')/(t+1)$.

Proposition 9 (p. 3, quoted). "For every $s,t\in\mathbb N^+$ and $s'\in\mathbb N$, the rooted tree $F:=T_{s,t,s'}$ is balanced if and only if $\rho_F\ge\max(s,s')$ and $\rho_F>1$, or equivalently $s'-1\le s\le t+s'$ and $(t,s')\ne(1,0)$."

Source. T. Jiang, Z. Jiang and J. Ma, Negligible obstructions and Turán exponents, arXiv:2007.02975v3 (30 January 2023), Proposition 9 and Figure 1 on p. 3; 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: the statement, Definitions 4 and 5 and Figure 1 were read on the page images of pp. 2--3.

Proof pointer

The paper prints no proof; it introduces the proposition by saying that the characterization is not hard (p. 3).

Dependencies

Definitions 4 and 5 (p. 2).

Bears on

• Problem 571: through Corollary 10, whose proof uses the proposition to show that the tree it builds is balanced.