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  "body": "\n## Definition\n\nLet $t, k, n$ be positive integers with $t \u003c k \u003c n$.\nA Steiner system $S(t, k, n)$ is a pair $(X, \\mathcal{B})$ where $X$ is a [[finite set|0BBTH3YSZA]] with $n$ elements.\nThe family $\\mathcal{B}$ is a set of $k$-element subsets of $X$, called blocks.\nThe condition is that each $t$-element subset of $X$ is contained in exactly one block.\n\n## Remarks\n\nEquivalently, a Steiner system $S(t, k, n)$ is a $t$-$(n, k, 1)$ [[block design|TGZATGBHTH]].\nCounting $t$-sets gives the number of blocks: $\\binom{n}{t} / \\binom{k}{t}$.\nExample: on the $9$ points of the grid $(\\Z/3\\Z)^2$, the $12$ lines of the affine plane form an $S(2, 3, 9)$.\nTwo distinct points $p, q$ lie on exactly one line, namely $\\set{p, q, -p - q}$.\nNon-example: there is no $S(2, 3, 8)$.\nIn an $S(2, 3, n)$, the blocks through a point split the other $n - 1$ points into pairs, so $n - 1$ must be even, and $8 - 1 = 7$ is odd.\nAn $S(2, 3, n)$ is a Steiner triple system; for $n \\ge 7$, one exists if and only if $n \\equiv 1$ or $3 \\pmod 6$.\nA [[finite projective plane|2HWB7P1DZN]] of order $q$ is an $S(2, q + 1, q^2 + q + 1)$.\n",
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