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Statement

Setting (pp. 252--253). The families $\mathscr B$, $\mathscr B_1$, $\mathscr B_2$, $\mathscr B'$, $\mathscr S$ and $\mathscr B_1'$, the graph $G'$ and the parameters $\gamma$, $\beta$, $\delta$ with $|\mathscr B_1|=\gamma\nu$, $|\mathscr B_2|=\beta\nu$ and $|\mathscr B_1'|=\delta\nu$, as defined for Lemma 3, with $\nu=\nu(G)$ for the fixed graph $G$.

Lemma 4 (p. 253, quoted). "We have $\tau(G)\le(3+3\delta-\beta)\nu$."

Proof sketch

Proof on pp. 253--254. Take the edges of $\mathscr B_1$ and of $\mathscr B_1'$ together with the edges common to $E[\mathscr B]$ and $E[\mathscr B']$; the last set has at most $3|\mathscr B'|-\beta\nu$ edges, which gives the count. Since $\mathscr B'$ is maximal in $G'$, every triangle of $G'$ meets $E[\mathscr B']$. A triangle of $G'$ that misses the common edges cannot have two or three edges in $E[\mathscr B']\setminus E[\mathscr B]$: two would make it a type-$(\mathscr B,1)$ triangle, absent from $G'$, and three would make it disjoint from $\mathscr B$. So it lies in $\mathscr S$ and, by maximality of $\mathscr B_1'$, meets $E[\mathscr B_1']$.

Dependencies

None outside the paper: the maximality of $\mathscr B'$ and $\mathscr B_1'$ and the absence of type-$(\mathscr B,1)$ triangles in $G'$.

Bears on

• Problem 167: one of the four inequalities whose weighted sum proves Theorem 5, $\tau(G)\le\frac{66}{23}\nu(G)$, toward Tuza's conjecture $\tau(G)\le2\nu(G)$. The paper's closing remark (p. 254) proposes replacing its $3\delta$ term by $(3-\varepsilon)\delta$ through induction, with no printed proof; the 2026 preprint of Yi replaces the bound $3|\mathscr B_1'|$ for covering $\mathscr S$ by $\frac{11}4|\mathscr B_1'|$ for its claimed constant $\frac{63}{22}$ (Corollary 1). It settles nothing the problem page leaves open.