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Source. Section 5.5, printed p. 462 (published PDF).

Statement. The maximum of $\sum_{e\in E(G)}x_e$ under

$$x_e\ge0,\qquad \sum_{e\ni v}x_e\le1\quad(v\in V(G)),\qquad \sum_{e\in E(G[S])}x_e\le\frac{|S|-1}{2} \quad(|S|\ge3\text{ odd})$$

is $\nu(G)$, and is attained by a zero-one matching indicator.

Proof. The indicator of any matching satisfies all these inequalities. Choose a minimum odd-set cover of capacity $\nu(G)$ using matching duality. For each singleton member use its vertex inequality; for every larger odd member use its odd-set inequality. Sum these inequalities. Each edge variable is counted at least once because the family covers all edges, and all variables are nonnegative. Hence

$$\sum_e x_e \le \sum_{S\in\mathcal S} \sum_{\text{edges covered by }S}x_e \le \sum_{S\in\mathcal S}c(S)=\nu(G).$$

A maximum matching indicator attains this bound. The empty graph has the empty feasible vector and objective zero. $\square$

The source's first paragraph prints the vertex sum as “less than one”. The required weak inequality is forced by its matching definition: a single selected edge has endpoint sums equal to one. The correction above is a compilation-supplied wording repair.

The source also announces exactness for every real weighted objective and equality with the full matching polytope. It explicitly postpones that stronger result to a separate paper. Unweighted exactness alone is not its proof.