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  "body": "\n## Description\n\nThe Rado graph $R$ has vertex set $\\N = \\set{0, 1, 2, \\dots}$.\nFor $i \u003c j$, the vertices $i$ and $j$ are adjacent when the binary digit of $j$ at position $i$ is $1$.\nEquivalently, $\\floor{j / 2^i}$ is odd.\n\nIt has the extension property, stated as follows.\nFor any two disjoint finite sets $U, V \\subseteq \\N$, some vertex $z \\notin U \\cup V$ is adjacent to every vertex of $U$ and to no vertex of $V$.\nCheck: let $N$ be larger than every element of $U \\cup V$ and put $z = 2^N + \\sum_{u \\in U} 2^u$.\nThen $z \u003e N$, so $z \\notin U \\cup V$, and every $w \\in U \\cup V$ satisfies $w \u003c z$.\nThe binary digit of $z$ at position $w$ is $1$ if $w \\in U$ and $0$ if $w \\in V$.\nSo $z$ is adjacent to every vertex of $U$ and to no vertex of $V$.\n\n## Remarks\n\nAny two countable [[graphs|00C0BR7D7H]] with the extension property are [[isomorphic|0F7QKGJS6B]], by a back-and-forth argument.\nSo the Rado graph is the [[Fraïssé limit|HJVRBDQSGG]] of the class of finite graphs.\nEvery finite graph is an induced subgraph of it, and every isomorphism between finite induced subgraphs extends to an automorphism.\nA countable [[random graph|TNT3DG0HRJ]] in which each edge is present independently with probability $1/2$ is isomorphic to $R$ with probability $1$.\nFor this reason $R$ is also called the countable random graph.\nThe first-order theory of $R$ is $\\aleph_0$-categorical.\n\n## Explanations\n\n### L1 human\n\nThe Rado graph is an infinite graph that contains every finite and every countable graph as an induced subgraph.\nIts key feature is that any finite wish list \"be joined to these vertices and not to those\" is met by some vertex.\nIf one draws a countable graph by flipping a fair coin for each pair of vertices, one gets this same graph with probability one.\n",
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