Let $t \lt n-1$,

A family { $V_1, V_2, ..., V_n$ } sub-spaces of an $n$-dimensional vector space $V$ is called $t$-**feasible** if it satisfies conditions (i) and (ii) below:

(i) $\dim(V_i) = t$, for all $i$ $(1 \leq i \leq n)$

(ii) for any $k$ $(2 \leq k \leq n)$, if $1 \leq i_1 \lt i_2 \lt ...\lt i_k \leq n$ , then $\dim(\cap_{j=1}^kV_{i_j}) \le n-k$

**Question:**

Given a $t$-**feasible** family { $V_1, V_2, ..., V_n$ } and a basis $B= ${$v_1, v_2, ..., v_n$} of $V$, is there a permutation $\pi \in S_n$ so that the family
{ span$(V_1 \cup v_{\pi(1)})$ , span$(V_2 \cup v_{\pi(2)})$, ...,span$(V_n \cup v_{\pi(n)})$ } is $(t+1)$-**feasible**?

Note:

Using Hall's Marriage theorem, one could prove that there is a permutation $\pi \in S_n$ so that the new family satisfy condition (i). The difficulty is in proving condition (ii). Specifically, when increasing the dimension of each sub-space by $1$, there is no guarantee that the dimension of some intersection will not increase by more than $1$.