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Graph Generated Union-Closed Families of Sets

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Graph Generated Unionclosed Families of Sets Emanuel Knill Los Alamos National Lab oratory Los Alamos NM knilllanlgov revised June Abstract S e j E Let G b e a graph with vertices V and edges E Let F f eE E g b e the unionclosed family of sets generated by E Then F is the family of subsets of V without isolated p oints Theorem There is an edge e E 1 such that fU F j U eg F This is equivalent to the following 2 assertion If H is a unionclosed family generated by a family of sets of maximum degree two then there is an x such that fU H j x U g 1 H This is a sp ecial case of the unionclosed sets conjecture To put this 2 result in p ersp ective a brief overview of research on the unionclosed sets conjecture is given A pro of of a strong version of the theorem on graph generated families of sets is presented This pro of dep ends on an analysis of the lo cal prop erties of F and an application of Kleitmans lemma Much of the pro of applies to arbitrary unionclosed families and can b e used to obtain b ounds on fU F j U eg F Preliminaries Unless stated otherwise all sets are assumed to b e nite Let N b e the set of nonnegative integers Dene n fi N j i ng and m n fi N j m i ng Let X b e a set A family of sets on X is a subset F of X the p ower set of X F is unionclosed intersectionclosed i for every U V F U V F U V F The family F is a p oset where the memb ers of F are ordered by inclusion For Y X let F fU F j U Y g Y b e the family induced by F in Y let F fU F j U Y g Y b e the family induced by F above Y let F fU Y j U F g Y b e the restriction of F to Y and let F fU n Y j U F g nY b e the restriction of F away from Y If Y fxg is a singleton the set brackets are omitted The degree of Y in F is dened as d Y F For families F Y of sets F and G let F G fU V j U F V G g and F G fU V j U F V G g The transpose or dual of F is the collection of sets F on F dened by hF j x X i F x where the F are counted with multiplicities A collection of sets is called x simple if each memb er o ccurs only once ie if it is a family of sets If the dual of F is not simple then there are elements of X which are not separated by any memb er of F Call F primitive on X if the dual of F is simple and S F X Let P b e a p oset The dual P of P is P with the reverse ordering A map f from P to a p oset Q is orderpreserving i x y implies f x f y P P The set of orderpreserving maps from P to Q is denoted by Q Q is a p oset with the p ointwise ordering i for all x x x The element x P covers y P i x y and x z y implies z x or z y For A P let A fx P j x y for some y Ag P A fx P j x y for some y Ag P Thus A is the order ideal and A is the order lter generated by A Sub P P scripts are omitted if the p oset is clear from context To simplify notation for x P let x fxg and x fxg If P is closed under least upp er b ounds then P is a joinsemilattice with the least upp er b ound of x and y denoted by x y the join of x and y Similarly if P is closed under greatest lower b ounds then P is a meetsemilattice with the greatest lower b ound of x and y denoted by x y the meet of x and y A lattice is b oth a joinsemilattice and a meetsemilattice Recall that any nite joinsemilattice can b e made into a lattice by adding a least element if necessary and similarly for meetsemilattices The meet join of elements in a joinsemilattice meetsemilattice is understo o d to b e the meet join in this extension If L is a lattice and x L satises that u v x implies u x or v x then x is joinirreducible Every element of L can b e represented as the join of the joinirreducible elements b elow it The set of joinirreducible elements excluding the least element is denoted by J L Meetirreducible elements are dened dually M L denotes the set of meetirreducible elements of L excluding the largest element There are canonical corresp ondences b etween nite semilattices and primitive union or intersectionclosed families of sets Clearly every union or intersectionclosed family considered as a p oset is a semilattice Let L b e a meetsemilattice Let F L fx J L j x Lg L Then F is a primitive intersectionclosed family of sets If L is a joinsemilattice then the corresp onding unionclosed family of sets consists of the family of complements in M L of memb ers of F L S Let F b e a primitive unionclosed family of sets on X F The S meetirreducible elements of F are given by the sets M F The x X nfxg union generators of F are given by GF J F fg This is a family of sets with the prop erty that no memb er is the union of any collection of the other memb ers If GF G dene F G F A family of sets G is a graph i for every U G U The memb ers of G are referred to as edges The graph G is simple i for every U G U A graph generated unionclosed family of sets satises that GF is a graph Unionclosed families of sets The following problem has b een attributed to PFrankl Duus Stanley Problem The unionclosed sets conjecture Let F be a unionclosed family of sets with at least one nonempty set Does there always exist an ele F ment x such that d x F The family of sets consisting of only the empty set do es not satisfy the assertion of this problem Henceforth all families of sets and semilattices are assumed to have at least two memb ers It is straightforward to check that the n unionclosed conjecture holds for F if n Using duality and the corresp ondence b etween unionclosed families of sets and joinsemilattices one arrives at other questions equivalent to the union closed sets conjecture The rst such question is obtained by complementing the sets of a unionclosed family This yields an intersectionclosed family order isomorphic to the p oset dual Problem Let F be an intersectionclosed family of sets Does there always S F exist an element x F such that d x F The second equivalent question is obtained from the rst by using the cor resp ondence b etween intersectionclosed families and meetsemilattices and by observing that if F is a primitive intersectionclosed family then the elements S of F are in onetoone corresp ondence with J F Problem Let L be a meetsemilattice Does there always exist a member x J L such that x L There is a corresp onding version for joinsemilattices The last equivalent ques tion to b e given here is obtained by considering a unionclosed family of sets with as a meetsemilattice Problem Let F be a unionclosed family of sets with F Is there always a member U of GF such that F F U The unionclosed sets conjecture can b e generalized as follows Knill Let P b e a p oset and let p b e the numb er of lters of P A meetsemilattice L has P P x L p the P density property i there exists x J L such that The quantity on the left of this inequality is called the P density of x in L The density of x is the density of x The density property is the density prop erty Problem The P density problem What meet semilattices have the P density property For P this reduces to Problem Ie L has the density prop erty i it satises the assertion of Problem If L has the density prop erty with witness a J L then we say that L a has the density prop erty Again the trivial example the oneelement semilattice do es not have the P density prop erty Some results related to the unionclosed sets conjecture are given next Recall that any intersectionclosed family is a meetsemilattice The next two results have b een rediscovered a few times and are given in S Theorem If the intersectionclosed family F contains F n fx y g for S some x y F then F has the density property P Let S F U U F F of the members of the intersection Theorem If the average size S F S closed family F is at most then F has the density property F Theorems and are obtained by double counting They can b e generalized substantially a long overdue analysis of the technique is given by Po onen These ideas can b e used to establish lower b ounds on counter examples to the unionclosed sets conjecture currently these b ounds are ab out S F F and The motivation underlying the work presented in Section is related to the ideas in One of the main results of Theorem characterizes union closed families F which satisfy that for every unionclosed extension G F there S G Here we consider extensions G of F in exists an x F with d x G the context of Problem In terms of the unionclosed sets conjecture this can b e seen to involve mo difying F by adding new generators and adding new elements to the generators of F S For a given unionclosed family F with F X dene S F sF X F X S then F satises the unionclosed sets As in Theorem if sF F X conjecture Wojcik studies the prop erties of s min fsF X j m S log m 2 m F X g He conjectures that s o as m In fact m m this is one of very few so far unproven apparently proper consequences of the unionclosed sets conjecture Wojcik also gives an exact version of his conjecture which follows from a slight extension of the unionclosed sets conjecture Wojciks conjecture can b e strengthened as follows unpublished For P ln i each n let n bn b e the binary expansion of n where n and i i l n or bn Let ln U n fi j i l n and bn or i l n and bn g i i The function U is a bijection from N to nite subsets of N It induces an ordering on such sets which is closely related to the lexicographic orderings If k l and U i U k U l then i l It follows that the families U n fU U U n g are unionclosed Let t minfS F j F
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