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Yet another generalization of the Kruskal-Katona theorem

G.F. Clements
1998 Discrete Mathematics  
An algorithm is given for calculating min labial, where the minimum is taken over all m-element subsets .~ of ~. If tl = t2 ..... t, = 1, it reduces to the Kruskal-Katona algorithm. (~)  ...  .~¢ of Tt, Ab.~ denotes the elements of Tt-b which precede at least one element of s~¢.  ...  This can be done using the Kruskal-Katona algorithm (1). Proof of the theorem We begin by giving an inductive formulation of Engel's order.  ... 
doi:10.1016/s0012-365x(97)00161-1 fatcat:tqv3hhb7tje5rjiuhcly7aqnxu

On γ-Vectors Satisfying the Kruskal–Katona Inequalities

Eran Nevo, T. Kyle Petersen
2010 Discrete & Computational Geometry  
We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal-Katona inequalities.  ...  In another direction, we show that if a flag (d-1)-sphere has at most 2d+2 vertices its γ-vector satisfies the Kruskal-Katona inequalities.  ...  There are many examples of connected chordal building sets B for which the hypotheses of Lemma 3.5 do not apply, and yet we still believe that the γ-vectors of the related nestohedra satisfy the Kruskal-Katona  ... 
doi:10.1007/s00454-010-9243-6 fatcat:3gcldtgcjzb2bcsdbabuj5rrs4

Author index to volume 184 (1998)

1998 Discrete Mathematics  
Huang, A study of the total chromatic number of equibipartite graphs (1-3) 49-60 Chen, X., Some families of chromatically unique bipartite graphs (Note) , G.F., Yet another generalization of the Kruskal-Katona  ...  theorem (l-3) 61 70 Colbourn, C.J., see R.  ... 
doi:10.1016/s0012-365x(97)83859-9 fatcat:lyk7pw4tsjgt7cisllruhse7ni

Odd and even hamming spheres also have minimum boundary

Janos Körner, Victor K. Wei
1984 Discrete Mathematics  
Sets with more general extremal properties of this kind yield good error-correcting codes for multi-terminal channels.  ...  The Hamming dkcunce of any two binary sequences is the number of positions in which they differ.  ...  (For various proofs of the Kruska- Katona theorem, cf. Kruskal [9& KaT 3~ [II, .) In order to quote Kruskal-Katona S, theorem, observe fhst that ti K.  ... 
doi:10.1016/0012-365x(84)90068-2 fatcat:tzqlig6m7rbwxocwfhvobgsd4y

Page 6776 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
F. (1-CO; Boulder, CO) Yet another generalization of the Kruskal-Katona theorem. (English summary) Discrete Math. 184 (1998), no. 1-3, 61-70.  ...  If t} = t2 =---=1t, = 1, it reduces to the Kruskal-Katona algorithm.” 98k:06005 06A07 05C70 Fleiner, Tamas (NL-MATH; Amsterdam) Covering a symmetric poset by symmetric chains.  ... 

Towards bounding sequential patterns

Chedy Raïssi, Jian Pei
2011 Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '11  
Unfortunately, Leck [13] affirmed that there is no theorem of the Kruskal-Katona type for sequential patterns.  ...  The next theorem, the Kruskal-Katona theorem, gives a tighter upper bound on the size of the shade for a set of k-subsets. Definition 6 (Shade). An itemset X ⊆ I is a k- subset if |X| = k.  ... 
doi:10.1145/2020408.2020612 dblp:conf/kdd/RaissiP11 fatcat:piy2qcl23reqnnh7w2ma24jkou

Book Review: Combinatorics of finite sets

Douglas B. West
1988 Bulletin of the American Mathematical Society  
Among the applications of the Kruskal-Katona Theorem is the Erdös-Ko-Rado Theorem, which itself has many interesting proofs and extensions.  ...  The Kruskal-Katona Theorem answers the question of how to choose m fc-element subsets of a set to minimize the number of k -1-element subsets contained in one or more of them.  ... 
doi:10.1090/s0273-0979-1988-15725-4 fatcat:zuam77ri4bbwpjqlhpiujuvp7q

Page 1747 of Mathematical Reviews Vol. 50, Issue 6 [page]

1975 Mathematical Reviews  
The author’s proof requires no background material other than Hall’s theorem itself and thus provides yet another illustration of the theme of “‘self-refinement” of Hall’s criterion expounded by the author  ...  The author gives a new proof of the generalization of P. Hall’s theorem [J. London Math. Soc. 10 (1935), 26-30; Zbl 10, 345] due to A. J. Hoffman and H. W.  ... 

Shadows and intersections: Stability and new proofs

Peter Keevash
2008 Advances in Mathematics  
We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\'asz.  ...  We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation  ...  Acknowledgments The author thanks Dhruv Mubayi for helpful conversations, Benny Sudakov for bringing references [1] and [14] to his attention, and an anonymous referee for their careful reading of  ... 
doi:10.1016/j.aim.2008.03.023 fatcat:iy6aes32mzdungiusbtyrtz4g4

Page 2793 of Mathematical Reviews Vol. , Issue 88f [page]

1988 Mathematical Reviews  
The proof of Theorem 2 makes use of the Kruskal-Katona simplicial complexes that minimize the number of j-simplices, j <i, among all simplicial complexes having p i-simplices.  ...  Swart [Quaestiones Math. 7 (1984), no. 2, 161-178; MR 85k:05080] yet another generalization—a;, b;-colouring—of graph colouring is introduced. a;, b;-colourability and an a;,b;-chromatic number of G, denoted  ... 

Some New Results on Macaulay Posets [chapter]

Sergei L. Bezrukov, Uwe Leck
2000 Numbers, Information and Complexity  
Macaulay posets are posets for which there is an analogue of the classical Kruskal-Katona theorem for finite sets.  ...  Introduction Macaulay posets are, informally speaking, posets for which an analogue of the classical Kruskal-Katona theorem for finite sets holds.  ...  Corollary 1 (Colored Kruskal-Katona theorem Corollary 2 The colored complexes are additive. The following theorem is the result of yet another application of the Kleitman's idea mentioned above.  ... 
doi:10.1007/978-1-4757-6048-4_6 fatcat:o6xl5kzztvdg7np2wl7lrz3jhe

Are there Hard Examples for Frege Systems? [chapter]

Maria Luisa Bonet, Samuel R. Buss, Toniann Pitassi
1995 Feasible Mathematics II  
It is shown that Bondy's theorem and a version of the Kruskal-Katona theorem actually have polynomial-size Frege proofs.  ...  on a theorem of Frankl.  ...  Acknowledgements We would like to thank the numerous people who have provided stimulating suggestions and input to this line of investigation.  ... 
doi:10.1007/978-1-4612-2566-9_3 fatcat:4e2z5jonanfcnj4rqtcwoddjky

Gotzmann Theorems for Exterior Algebras and Combinatorics

Annetta Aramova, Jürgen Herzog, Takayuki Hibi
1997 Journal of Algebra  
Yet another formulation of the KruskalKatona theorem, which really points to the core of the theorem, is the following: Let I ; E be a graded ideal. Then ␤ I F ␤ I lex Ž . Ž . 0j 0j for all j.  ...  We will generalize this result and prove the following ''higher'' KruskalKatona theorem: Ž .  ...  In general, for each subset w x w x w x of n , we write for the complement of in n , i.e., s n y . w x Given a simplicial complex ⌬ on the vertex set n , we may associate Ä 4 ⌬s :f⌬.  ... 
doi:10.1006/jabr.1996.6903 fatcat:om3xrfteerbi5pdihgvmjqvomq

Extremal Combinatorics: with Applications in Computer Science by Stasys Jukna, Springer, 2001, xvii + 375 pp. 32.50; $49.95, ISBN 3540663134

Imre Leader
2004 Combinatorics, probability & computing  
The book has a reasonable number of exercises, and would be easily accessible to a beginning research student.  ...  survey article of Babai [1] , while reconstruction is covered in greater depth by the surveys of Bondy [2] , Bondy and Hemminger [3] and Nash-Williams [6] .  ...  Perhaps the most striking is that there is no mention of the Kruskal-Katona theorem.  ... 
doi:10.1017/s096354830321244x fatcat:7gh6o3yplzbbpco26gowr57pjq

Page 1360 of Mathematical Reviews Vol. , Issue 83d [page]

1983 Mathematical Reviews  
A number of LYM-type inequalities are given for these posets and related structures, as well as inequalities related to generalizations of the Kruskal- Katona theorem.  ...  In part II, starting from a generalization of the binomial theorem, a development of Rota’s theory of polynomial sequences of binomial type to the case of countably many noncommuting variables is given  ... 
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