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  "body": "\n## Statement\n\nTwo players build a graph in rounds.\nIn each round, Builder adds one new vertex and joins it to any chosen set of earlier vertices.\nThen Assigner at once gives the new vertex a colour that differs from the colours of all its neighbours.\nA colour is never changed later, and Assigner sees only the part of the graph built so far.\nThis is an on-line [[colouring|0874138YP6]].\n\nLet $n$ be a positive integer.\nThen Builder has a strategy such that the final graph is a [[forest|09JP6G9FTA]] with at most $2^{n-1}$ vertices, and Assigner is forced to use $n$ distinct colours, whatever Assigner does.\n\n## Remarks\n\nA forest is bipartite, so it has an off-line colouring with $2$ colours.\nThe theorem shows that an on-line Assigner can be forced to use arbitrarily many colours even on forests.\n",
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  "name": "On-line colouring of forests needs few vertices",
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