This post is copied from math.SE in the following link:
https://math.stackexchange.com/questions/456398/a-conjecture-about-vector-space
I have posted the question two days ago, but receive no answer yet. Therefore I think maybe I can post it on MO and invite more attention. If this violates some policy, let me know and I'll delete the post.
I am a college lecturer in Macau, and this conjecture is based on some discussions with my colleagues. We think this problem can be set for some math competitions for college students, however we have not reached conclusion regarding this conjecture. Therefore I post it out and invite your attention. My description of the conjecture using English may not look professional, and I welcome your editing to make it more sound. Thank you very much.
Let $V$ be a $(r+1)$-dimensional vector space, and $p$ be a positive integer and $1\leq p\leq r-1$. Let $$X=\{v_1,\cdots,v_{2r+1-p}\}\subseteq V$$ be a finite set containing $(2r+1-p)$ different vectors, and $\{u,v\}$ is a linearly-independent set for any $u,v\in X$ such that $u\neq v$. Moreover, for any set of $(2s+2−p)$ vectors in $X$ there exists no $(s+1)$-dimensional subspace that contains said set, where $s=p,p+1,\cdots,r-1$.
Prove or disprove the following conjecture: $X$ can be divided into two non-intersecting non-empty subsets $$X=X_1\cup X_2$$ such that $X_1$ consists of $(r+1)$ linearly-independent vectors and $X_2$ consists of $(r-p)$ linearly-independent vectors.
