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Statement

Setting (pp. 252--253). As for Lemma 2: $G$ a fixed graph, $\nu=\nu(G)$, $\mathscr B$ a fixed independent family of $\nu$ triangles, $E[\cdot]$ the edges of the triangles of a family, $\mathscr B_1$ with $|\mathscr B_1|=\gamma\nu$, the graph $G'$ obtained by deleting $E[\mathscr B_1]$, and $\mathscr B_2$ with $|\mathscr B_2|=\beta\nu$. On p. 253: let $\mathscr B'$ be an independent family of triangles in $G'$ of maximum size subject to $|E[\mathscr B']\setminus E[\mathscr B]|\ge\beta\nu$ (such a family exists because of $\mathscr B_2$), so that $|\mathscr B'|\le\nu(1-\gamma)$. Let $\mathscr S$ be the set of triangles $T$ that have exactly one edge in common with $E[\mathscr B']$, that edge lying in $E[\mathscr B']\setminus E[\mathscr B]$. Let $\mathscr B_1'$ be an independent subset of $\mathscr S$ of maximum size, and define $\delta$ by $|\mathscr B_1'|=\delta\nu$. The definition of $\mathscr S$ does not name the graph; the proofs of Lemmas 3 and 4 use it for triangles of $G'$.

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

Proof sketch

Proof on p. 253. Run the construction of Lemma 1 inside $G'$, with $\mathscr B'$ in place of $\mathscr B$ and $\mathscr B_1'$ in place of $\mathscr B_1$. The new point is that a triangle of $\mathscr B'$ paired with $T_1\in\mathscr B_1'$ has the shared edge as its only edge outside $E[\mathscr B]$, since otherwise it would be a type-$(\mathscr B,1)$ triangle, and $G'$ has none; so swapping members of pairs keeps the constraint $|E[\cdot]\setminus E[\mathscr B]|\ge\beta\nu$, and the maximality of $\mathscr B'$ applies as before. This gives a transversal of $G'$ with at most $3|\mathscr B'|-\delta\nu$ edges; adding $E[\mathscr B_1]$, of size $3\gamma\nu$, gives a transversal of $G$ of size at most $(3-\delta)\nu$.

Dependencies

Lemma 1's construction, rerun 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 $\tau(G)\le2\nu(G)$; it is also one of the four lemmas that the 2026 preprint of Yi restates for its claimed constant $\frac{63}{22}$ (Corollary 1). It settles nothing the problem page leaves open.