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  "body": "\n## Definition\n\nLet $\\kappa$ be a [[cardinal|0EBACQFAZW]], and write $[\\kappa]^2$ for the set of two-element subsets of $\\kappa$.\nThe cardinal $\\kappa$ is *weakly compact* if:\n\n1. $\\kappa$ is [[uncountable|0W7FBYS1F1]].\n2. For every function $c \\colon [\\kappa]^2 \\to \\set{0, 1}$ there is a set $H \\subseteq \\kappa$ with $\\abs{H} = \\kappa$ such that $c$ is constant on $[H]^2$.\n\nClause 2 is written $\\kappa \\to (\\kappa)^2_2$.\n\n## Remarks\n\nThe cardinal $\\aleph_0$ satisfies clause 2 by Ramsey's theorem, and it fails only clause 1.\nThe cardinal $\\aleph_1$ fails clause 2: Sierpiński's colouring compares the order of $\\omega_1$ with the order of an injective family of real numbers, and it has no uncountable homogeneous set.\nEquivalent form: $\\kappa$ is weakly compact if and only if $\\kappa$ is [[inaccessible|NW4DREQ5DQ]] and has the tree property.\nThe tree property says that every tree of height $\\kappa$ whose levels have size less than $\\kappa$ has a branch of length $\\kappa$.\nAnother equivalent form: $\\kappa$ is inaccessible, and a compactness theorem holds for the infinitary logic $\\mathcal{L}_{\\kappa, \\kappa}$.\nIt says that a set $\\Sigma$ of at most $\\kappa$ sentences has a model whenever every subset of $\\Sigma$ of size less than $\\kappa$ has one.\n",
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