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  "body": "\n## Definition\n\nLet $p, \\ell, \\gamma, \\pi$ be positive integers.\nA *configuration* $(p_\\gamma\\ \\ell_\\pi)$ is a finite [[incidence structure|FGKNAFNHA3]] of points and lines such that:\n\n1. there are $p$ points and $\\ell$ lines;\n2. each point lies on exactly $\\gamma$ lines;\n3. each line contains exactly $\\pi$ points;\n4. two distinct points lie on at most one common line.\n\n## Remarks\n\nClause 4 says that the structure is a [[partial linear space|CX07CWPCDX]]; it is equivalent to saying that two distinct lines meet in at most one point.\nCounting incident point-line pairs in two ways gives $p \\gamma = \\ell \\pi$.\nWhen $p = \\ell$, and hence $\\gamma = \\pi = k$, the configuration is *symmetric* and is written $(p_k)$.\nA *geometric* configuration is one whose points and lines are points and lines of the real (or another) projective plane, with incidence as in the plane.\n\nExample: the complete quadrilateral, made of $4$ lines in the real plane with no three concurrent and their $6$ intersection points, is a $(6_2\\ 4_3)$ configuration.\nEach line meets the other three lines in $3$ distinct points, each point lies on $2$ of the lines, and $6 \\cdot 2 = 4 \\cdot 3$.\n\nNon-example: a triangle together with one median has $4$ points and $4$ lines.\nThe vertex the median starts from lies on $3$ of the lines, but each other point lies on only $2$, so clause 2 fails.\n",
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