I'm a bit confused as to how the $K_3$ graph with $x(e) = \frac12$ for all $e \in E$ fails the third set of inequalities for Edmond's perfect matching polytope ($x(\delta(U)) \geq 1$ for each $U \subseteq V, |U| \geq 3, |U| \text{ is odd}$).
In this case only one set of vertices, $V$, meets the side condition for the inequalities and $x(\delta(U)) = \frac32$. I understand why $K_3$ can't have a perfect matching and all I just don't see how this third set of inequalities enforces this.
I know I must be missing something unbelievably simple but I can't for the life of me figure out what it is. Am I just misinterpreting what $x(\delta(U))$ means? I'm just taking it to be the sum of $x(e)$ for all $e$ incident to $U$.
I realized I misinterpreted $x(\delta(U))$ as summing all edges incident to $U$ while it actually only counts edges exiting $U$