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  "body": "\n## Statement\n\nLet $V$ be a [[vector space|000AYPA9AJ]] of dimension $n \\ge 1$ over a field.\nLet $B_1, \\dots, B_n$ be [[bases|0KCQ9Y3QKA]] of $V$, regarded as $n$ disjoint labelled families of $n$ vectors.\nThen the $n^2$ vectors can be arranged in an $n \\times n$ grid with the following properties.\nFor each $i$, the $i$-th row consists of the vectors of $B_i$.\nEach column is a basis of $V$.\n\n## Remarks\n\nThe conjecture is due to Gian-Carlo Rota, and it is open.\nIt has a natural form for [[matroids|PX31ZNYS3Z]]: given $n$ [[bases|QTSQR7130G]] of a matroid of rank $n$, the elements can be arranged in such a grid with each column a basis.\nFor $n = 2$ the statement is easy to check by hand: with $B_1 = \\set{a, b}$ and $B_2 = \\set{c, d}$, at least one of the pairings $\\set{a, c}, \\set{b, d}$ or $\\set{a, d}, \\set{b, c}$ gives two bases.\nHuang and Rota related the vector space form to the Alon–Tarsi conjecture on Latin squares, which gives the conjecture for some even $n$ over fields of characteristic zero.\nAsymptotic versions are known that produce many disjoint column bases, but not $n$ of them in general.\n\n## Explanations\n\n### L1 human\n\nStart with $n$ bases of an $n$-dimensional space, each written as a row of $n$ vectors.\nThe question is whether one can reorder each row so that every column also becomes a basis.\nIt looks like a puzzle in the style of Latin squares, and it is still unsolved in general.\n",
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      "first_stated": "1989",
      "posed_by": "Gian-Carlo Rota"
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    "msc": [
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    "name": "Rota's basis conjecture",
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