# A conjecture about vector space (repost from math.SE)

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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.