Let $G$ be a planar triangulation on $3m$ edges and $m+2$ vertices. Let $A$ be the binary matrix obtained from the incidence matrix of $G$ by deleting a row (equivalently we require the rows of $A$ to form a basis of the cocycle space of $G$ over $GF(2)$).

My question is: What additional criteria must be assumed (if any) to guarantee there is an $(m−1)$-dimensional subspace of $GF(2)^{m+1}$ which does not contain any column of $A$?

It is easy to see that no $7$ columns are the nonzero words of a $3$-dimensional subspace, so it is not trivially blocked.

On the other hand, there are examples of $3m$ length $m+1$ words, not arising from a planar graph, which do not contain the nonzero points of a $3$-dimensional subspace and which block every $(m-1)$-dimensional subspace. (For example consider the set of weight $2$ words in $GF(2)^5$ along with $10000$ and $01000$ - corresponding to $K_5$ with an additional vertex adjacent to exactly two vertices of $K_5$.)

Any known bounds on the size of a skew set of this type other than that of Bose and Burton would be helpful. Relevant references would be much appreciated.