{"id":23199,"date":"2013-05-29T23:55:06","date_gmt":"2013-05-30T03:55:06","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=23199"},"modified":"2014-07-09T15:04:00","modified_gmt":"2014-07-09T20:04:00","slug":"union-closed-buckets","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/05\/29\/union-closed-buckets\/","title":{"rendered":"Union-Closed Buckets"},"content":{"rendered":"

\"Buckets-3-color\"<\/a>In the mines of mathematical conundrums there are a few gems that shine\u00a0brightly.\u00a0\u00a0 Such problems are illuminated with the delight of recreational enjoyment.\u00a0\u00a0\u00a0 Whether it be folding unique shapes with paper or exploring the elementary nature of prime numbers, these problems draw a general audience into the awe of patterns and relationships.\u00a0 Over the years, many communicators of these gems have looked for ways to make them shine even brighter.<\/p>\n

<\/p>\n

The greatest communicator of mathematical conundrums in modern history was Martin Gardner.\u00a0 Gardner wrote the monthly\u00a0Mathematical Games<\/i>\u00a0column for\u00a0Scientific American\u00a0<\/i>from 1956 to 1981. His first article was on a unique way to fold paper and its properties entitled\u00a0<\/i>\u201cHexaflexagons\u201d.<\/i>\u00a0 \u00a0In the same vein with his original article, Gardner uniquely picked the right gems and polished them to shine the brightest before being displayed.\u00a0 Following his example, I would like to recast a series of abstract problems into problems of a more recreational nature. To start the series off, I have chosen an unsolved problem in Set Theory known as the Union-Closed Sets Conjecture proposed by P\u00e9ter Frankl in 1979.<\/p>\n

Imagine a shelf with finitely many buckets on it.\u00a0\u00a0 Each bucket is either empty or contains at least one\u00a0colored ball (let most of the buckets have more than one) with the condition that\u00a0the\u00a0buckets\u00a0cannot all\u00a0be empty.\u00a0\u00a0 Furthermore,\u00a0 let it be true that I can take any two buckets off\u00a0 the shelf and empty their contents into one bucket not on the shelf (call it bucket B) such that bucket B, with its new contents, is identical to some bucket on the shelf.\u00a0\u00a0 Thus, do at least half of the buckets on the shelf share a ball of the same color?<\/p>\n

The formal generalized statement of the conjecture is the following:<\/p>\n

Union-Closed Sets Conjecture:<\/b>\u00a0(P\u00e9ter Frankl , 1979)\u00a0 If\u00a0 the union of any two sets in a finite family of finite sets belongs to the family, then there exists an element of the sets that exists in at least half of the sets.<\/p>\n

There are several known results which can easily\u00a0be\u00a0found by a quick search.\u00a0 For instance, Ian Roberts and Jamie Simpson in their paper \u201cA note on the union-closed sets conjecture\u201d in 2010 proved that the conjecture is true for families with at most 46 sets (or 46 buckets to keep with the recreational statement of the conjecture).<\/p>\n

However, the general case is still very much unsolved. \u00a0It is not clear on where a solution of such a problem can come from. \u00a0Sometimes it can come from readers like you. \u00a0 Can you provide a solution?<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

In the mines of mathematical conundrums there are a few gems that shine\u00a0brightly.\u00a0\u00a0 Such problems are illuminated with the delight of recreational enjoyment.\u00a0\u00a0\u00a0 Whether it be folding unique shapes with paper or exploring the elementary nature of prime numbers, these … Continue reading →<\/span><\/a><\/p>\n

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