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  "body": "\n## Definition\n\nLet $\\Pi$ be a [[projective plane|DANFNN8C4Y]].\nA *blocking set* of $\\Pi$ is a set $B$ of points such that every line of $\\Pi$ contains a point of $B$.\nA blocking set is *nontrivial* if it contains no line of $\\Pi$.\n\n## Remarks\n\nSome authors include \"contains no line\" in the definition of a blocking set.\n\nBruen's bound: a nontrivial blocking set in a plane of [[order|K2YVMTGTBB]] $n$ has at least $n + \\sqrt{n} + 1$ points, with equality exactly for the point set of a Baer subplane.\n\nExample: let $q$ be a prime power and view the [[plane $\\mathrm{PG}(2, q)$|CSMB49BDPT]] as a subplane of $\\mathrm{PG}(2, q^2)$.\nIts $q^2 + q + 1$ points form a nontrivial blocking set of $\\mathrm{PG}(2, q^2)$.\nCheck: each point of the subplane lies on $q^2 + 1$ lines of the large plane, of which $q + 1$ are lines of the subplane.\nA line of the large plane through two points of the subplane is a line of the subplane, so the other $q^2 - q$ lines meet the subplane in one point.\nThis gives $(q^2 + q + 1)(q^2 - q) = q^4 - q$ such lines, and with the $q^2 + q + 1$ lines of the subplane that is $q^4 + q^2 + 1$, all lines of $\\mathrm{PG}(2, q^2)$.\nNo line of the large plane lies inside the subplane, since it has $q^2 + 1 \u003e q + 1$ points.\n\nNon-example: the point set of a line $\\ell$ meets every line, but it contains $\\ell$, so it is a trivial blocking set.\n",
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