matroid theory - Representability of polymatroids over $GF(2)$

Here's a construction due to Stefan van Zwam.

Let $U_{2,4}$ be a 4-point line with ground set $[4]$, and rank function $r$. For each $A subset [4]$, let $chi(A)$ be 1, if $1 in A$, and 0 if $1 notin A$. For each $k in mathbb{N}$, let $S_k$ be the polymatroid with ground set $[4]$ and rank function $r+kchi$.

Lemma. $S_k$ is not representable over $mathbb{F}_2$, for every $k$.

Proof. Let $(V_i : i in [4])$ be a representation of $S_k$ over $mathbb{F}_2$. Choose a basis $B_i$ for each $V_i$. Since {1} has rank $k+1$ and {1,2,3,4} has rank $k+2$, we may assume that $B_1$ consists of the first $k+1$ standard basis vectors in $mathbb{F}_2^{k+2}$. Now, since {1,2}, {1,3}, and {1,4} all have rank $k+2$, it follows that $B_2, B_3$, and $B_4$ all must have last coordinate equal to 1. In particular, $B_2 + B_3 +B_4 neq 0$. Also, since every two element subset of {2,3,4} has rank 2, it follows that $B_2, B_3$ and $B_4$ are distinct. But now {2,3,4} must have rank 3 in $S_k$, a contradiction. $square$

Moreover, it is easy to check that every minor of $S_k$ is representable over $mathbb{F}_2$, where minors of polymatroids are defined in the obvious way. Therefore, the set of binary polymatroids has an infinite set of excluded minors.