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  "body": "\n## Statement\n\nLet $v, k, \\lambda$ be positive integers with $k \u003c v$.\nLet $(X, \\mathcal{B})$ be a $2$-$(v, k, \\lambda)$ [[block design|TGZATGBHTH]] with $b$ blocks.\nThen $b \\ge v$.\n\n## Remarks\n\nProof idea: let $r$ be the number of blocks through a point, and let $N$ be the $v \\times b$ [[incidence matrix|9DNFN8M83X]] of points against blocks.\nThen $N N^{T} = (r - \\lambda) I + \\lambda J$, where $I$ is the identity matrix and $J$ is the all-ones matrix of size $v$.\nIts [[determinant|0C3CFHDN6R]] is $rk (r - \\lambda)^{v - 1}$.\nFrom $r(k - 1) = \\lambda(v - 1)$ and $k \u003c v$ one gets $r \u003e \\lambda$, so the determinant is not zero.\nSo $N$ has rank $v$, and a $v \\times b$ matrix of rank $v$ needs $b \\ge v$.\nExample: the [[Fano plane|QQ8GW1ASZM]] is a $2$-$(7, 3, 1)$ design with $b = v = 7$, so the bound is sharp.\nThe designs with $b = v$ are the symmetric designs; the [[Bruck–Ryser–Chowla theorem|R43K0GTTDS]] restricts their parameters.\nThe hypothesis $k \u003c v$ is needed.\nIf $k = v$, every block is the whole set $X$, and $\\lambda$ copies of $X$ form a design with $b = \\lambda$ blocks, which can be less than $v$.\n\n## Explanations\n\n### L1 human\n\nIn a balanced design where every two points share the same number of blocks, there are at least as many blocks as points.\nSo one cannot save on blocks: balance costs at least one block per point.\n\n### L2 human\n\nFisher's inequality is a necessary condition for the existence of a design, beyond the divisibility conditions $bk = vr$ and $r(k-1) = \\lambda(v-1)$.\nIts proof by the rank of the incidence matrix is a model of the [[linear algebra method|DMYGZ3YZ6V]] in combinatorics.\nThe equality case $b = v$ leads to symmetric designs, such as [[finite projective planes|2HWB7P1DZN]] and the designs built from [[difference sets|MG1KM4S1BV]].\n",
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