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Yet another generalization of the Kruskal-Katona theorem

1998
*
Discrete Mathematics
*

An algorithm is given for calculating min labial, where

doi:10.1016/s0012-365x(97)00161-1
fatcat:tqv3hhb7tje5rjiuhcly7aqnxu
*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. ...##
###
On γ-Vectors Satisfying the Kruskal–Katona Inequalities

2010
*
Discrete & Computational Geometry
*

We present examples

doi:10.1007/s00454-010-9243-6
fatcat:3gcldtgcjzb2bcsdbabuj5rrs4
*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*...##
###
Author index to volume 184 (1998)

1998
*
Discrete Mathematics
*

Huang, A study

doi:10.1016/s0012-365x(97)83859-9
fatcat:lyk7pw4tsjgt7cisllruhse7ni
*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. ...##
###
Odd and even hamming spheres also have minimum boundary

1984
*
Discrete Mathematics
*

Sets with more

doi:10.1016/0012-365x(84)90068-2
fatcat:tzqlig6m7rbwxocwfhvobgsd4y
*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. ...##
###
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

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

doi:10.1145/2020408.2020612
dblp:conf/kdd/RaissiP11
fatcat:piy2qcl23reqnnh7w2ma24jkou
*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. ...##
###
Book Review: Combinatorics of finite sets

1988
*
Bulletin of the American Mathematical Society
*

Among

doi:10.1090/s0273-0979-1988-15725-4
fatcat:zuam77ri4bbwpjqlhpiujuvp7q
*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. ...##
###
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

2008
*
Advances in Mathematics
*

We give a short new proof

doi:10.1016/j.aim.2008.03.023
fatcat:iy6aes32mzdungiusbtyrtz4g4
*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*...##
###
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]

2000
*
Numbers, Information and Complexity
*

Macaulay posets are posets for which there is an analogue

doi:10.1007/978-1-4757-6048-4_6
fatcat:o6xl5kzztvdg7np2wl7lrz3jhe
*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. ...##
###
Are there Hard Examples for Frege Systems?
[chapter]

1995
*
Feasible Mathematics II
*

It is shown that Bondy's

doi:10.1007/978-1-4612-2566-9_3
fatcat:4e2z5jonanfcnj4rqtcwoddjky
*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. ...##
###
Gotzmann Theorems for Exterior Algebras and Combinatorics

1997
*
Journal of Algebra
*

*Yet*

*another*formulation

*of*

*the*

*Kruskal*᎐

*Katona*

*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''

*Kruskal*᎐

*Katona*

*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⌬. ...

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

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*. ...

##
###
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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