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We prove the following colorful Helly-type result: Fix k d-1. Assume A₁, , A₃+ (₃-₊) +₁ are finite sets (colors) of nonzero vectors in ᵈ. If for every rainbow sub-selection R from these sets of size at most \d+1, 2 (d-k+1) \, the system a, x 0, \; a R has at least k linearly independent solutions, then at least one of the systems a, x 0, \; a Aᵢ, i d+ (d-k) +1 has at least k linearly independent solutions. A rainbow sub-selection from several sets refers to choosing at most one element from each set (color). The Helly-number \d+1, 2 (d-k+1) \ and the number of colors d+ (d-k) +1 are optimal. Our key observation is a certain colorful Carath\'eodory-type result for positive bases.
Г. А. Иванов (Tue,) studied this question.