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  "body": "\n## Statement\n\nLet $d \\ge 1$ be an integer and let $\\alpha \\in (0, 1]$.\nThere is a number $\\beta = \\beta(d, \\alpha) \u003e 0$ with the following property.\nLet $\\mathcal{F}$ be a finite family of $n \\ge d + 1$ [[convex sets|02AS45T3H9]] in $\\R^d$.\nAssume that at least $\\alpha \\binom{n}{d+1}$ of the subfamilies of $\\mathcal{F}$ with $d + 1$ members have a common point.\nThen some point of $\\R^d$ lies in at least $\\beta n$ members of $\\mathcal{F}$.\n\n## Remarks\n\nThe theorem is due to Katchalski and Liu.\nOne can take $\\beta = 1 - (1 - \\alpha)^{1/(d+1)}$, a value found by Kalai, and this value is the best possible.\nA proof looks at the lexicographically smallest points of the nonempty intersections of $d$-element subfamilies and counts how often each such point is caught by the $(d+1)$-subfamilies.\nFor $\\alpha = 1$ every $d + 1$ members meet, and [[Helly's theorem|F1W3QRBNGJ]] gives a point in all $n$ members.\nThe constant $\\beta$ depends only on $d$ and $\\alpha$, not on $n$ or on the sets.\n\n## Explanations\n\n### L1 human\n\nHelly's theorem says: if every $d + 1$ convex sets of a family meet, then all of them meet.\nThe fractional version asks for less and gives less.\nIf only a fixed fraction of the $(d+1)$-tuples meet, then a fixed fraction of all the sets still share a point.\nSo \"many local meetings\" force \"one big meeting\".\n\n### L2 human\n\nThe fractional Helly theorem is the key step in the proof of the $(p, q)$-theorem of Alon and Kleitman.\nIt is a basic tool in the theory of geometric transversals.\nTogether with [[Helly's theorem|F1W3QRBNGJ]] and the [[centerpoint theorem|BWEGTCW5RJ]], it belongs to the core of combinatorial convexity.\n",
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