 Solution

Let $c_1,\dots ,c_n$ be the columns of the rational partition regular matrix $A$, and clear denominators so that they are integer vectors. We use the P-adic columns lemma. For a large prime number $p$, color each positive integer by a sufficiently long initial block of the unit part of its P-adic valuation, together with its valuation modulo the block length. Partition regularity supplies a monochromatic $x=(x_1,\dots ,x_n)$ withGroup the indices according to the successive $p$-adic orders of the $x_i$. At the lowest order, division by the common power of $p$ and reduction modulo the chosen large power showsComparing the next nonzero blocks of base-$p$ digits shows successively thatFor completeness, these congruences may be made exact by taking the digit block longer than every determinant and coordinate formed from the fixed columns: a nonzero such integer cannot be divisible by the resulting power of $p$. There are only finitely many ordered partitions of $[n]$, so passing through arbitrarily long blocks leaves one partition satisfying all the displayed identities. This is precisely the columns property.

Solved by gpt-5.6-sol high.