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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $X$ be a set with $n$ elements.\nLet $S_1, \\ldots, S_n$ be $n$ distinct subsets of $X$.\nThen there is an element $x \\in X$ such that the sets $S_1 \\setminus \\set{x}, \\ldots, S_n \\setminus \\set{x}$ are still distinct.\n\n## Remarks\n\nSuppose that no such $x$ exists.\nThen for each $x \\in X$ there are $i \\ne j$ with [[symmetric difference|0ME21QN125]] $S_i \\mathbin{\\triangle} S_j = \\set{x}$; join $S_i$ and $S_j$ by an edge labelled $x$.\nThis gives a graph on the $n$ sets with $n$ edges, all with distinct labels, so it contains a cycle.\nGoing around the cycle, the symmetric differences of the edges combine to $\\emptyset$, but a symmetric difference of distinct singletons is not empty, a contradiction.\n\nExample: $X = \\set{1, 2}$ with $S_1 = \\emptyset$ and $S_2 = \\set{1}$.\nRemoving $x = 2$ leaves $\\emptyset$ and $\\set{1}$, which are still distinct.\nThe bound is sharp: for $X = \\set{1, \\ldots, n}$ the $n + 1$ sets $\\emptyset, \\set{1}, \\ldots, \\set{n}$ become non-distinct after removing any element $x$, since $\\set{x}$ becomes $\\emptyset$.\n\n## Explanations\n\n### L1 human\n\nDescribe $n$ different objects by $n$ yes/no features.\nBondy's theorem says that one of the features is redundant: you can drop it and still tell all the objects apart.\nWith $n + 1$ objects this can fail.\n\n### L2 human\n\nThe theorem is a small result of extremal set theory with a short graph proof.\nIt says that $n$ distinct rows of a $0$–$1$ matrix with $n$ columns stay distinct after deleting a suitable column.\nIts proof is a typical use of a graph to encode a family of sets.\n",
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