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On k-homogeneous posets and graphs

1991
*
Journal of combinatorial theory. Series A
*

We show that if a countable

doi:10.1016/0097-3165(91)90018-c
fatcat:dozvkvszc5fitpfrfio5o2xnye
*poset*(52, < ) is l-*and*4-*homogeneous*, then it is*k*-*homogeneous*for each*k*E N. ... We also show that for every*k*E fV there are continuousiy many countable &categorical universal*graphs*which are*k*-*homogeneous*but not (*k*+ l)*homogeneous*; this answers a question of R. ... Here we examine the extent to which*k*-*homogeneity*implies (*k*+ l)-*homogeneity*or (*k*+ l)transitivity, particularly for infinite*posets**and**graphs*. ...##
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Page 05 of Mathematical Reviews Vol. , Issue 88b
[page]

1988
*
Mathematical Reviews
*

If a # b

*and*there is an ele- ment c < a,b, then a,b xk for every*k*€ mub{a, b}. Theorem 4: A finite modular 0, 1-*poset*is algebraically semimodular. ... More precisely, for every regular cardinal x*and*regular cardinals u,v <*k*, there are precisely 2" doubly*homogeneous*chains whose 05 COMBINATORICS 672 points have character (u,v) (the lengths of minimal ...##
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On polymorphism-homogeneous relational structures and their clones

2014
*
Algebra Universalis
*

Eventually, we completely characterize the countable polymorphism-

doi:10.1007/s00012-014-0310-3
fatcat:mx6av4chwjgupaoxxum5wzdqri
*homogeneous**graphs*, the polymorphism-*homogeneous**posets*of arbitrary size,*and*the countable polymorphism-*homogeneous*strict*posets*. ... We study polymorphism-*homogeneity*from a model-theoretic, an algebraic,*and*a combinatorial point of view. ... Let A be a countable set*and*let L be a meet-complete sublattice of E(A). Then L is polymorphism-*homogeneous*if*and*only if it is arithmetical. ...##
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More Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
[article]

2012
*
arXiv
*
pre-print

In this paper, we compute universal minimal flows of groups of automorphisms of uncountable ω-

arXiv:1204.0037v2
fatcat:vqbtrz5utfegvdyq63zcjs55qm
*homogeneous**graphs*, K_n-free*graphs*, hypergraphs, partially ordered sets,*and*their extensions with an ω-*homogeneous*... suggesting the present construction of ω*homogeneous*structures. ... I am also grateful to Lionel Nguyen Van Thé for careful reading*and*valuable suggestions. ...##
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Hereditary properties of combinatorial structures: posets and oriented graphs
[article]

2007
*
arXiv
*
pre-print

*posets*

*and*oriented

*graphs*. ... We also show that there is an initial jump in the possible unlabelled speeds of hereditary properties of

*posets*, tournaments

*and*directed

*graphs*, from bounded to linear speed,

*and*give a sharp lower bound ...

*graphs*of maximum degree at most

*one*,

*and*the property R consisting of all

*posets*with comparability

*graphs*of minimum degree n − 1. ...

##
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Page 1570 of Mathematical Reviews Vol. , Issue 2002C
[page]

2002
*
Mathematical Reviews
*

Under the mild technical assumption that a, 4 0

*and*a, = 0, the authors show that the tight*graphs*are precisely the 1-*homogeneous*distance-regular*graphs*. ... of*k*-chains [resp. antichains], then the sequences (A;,A2,...) amd (A;,A2,...) are weakly de- creasing*and*form conjugate partitions of the number N = | P|, it follows that to every finite*poset*P a ( ...##
###
Finite Homomorphism-Homogeneous Permutations via Edge Colourings of Chains

2012
*
Electronic Journal of Combinatorics
*

In this note, we characterise all permutations

doi:10.37236/2271
fatcat:gru737avtrgznhemoc7z2tvfgm
*on*a finite set enjoying this property. ... with loops) whose non-loop edges are coloured in two colours, thereby 'splitting' the linear order into two*posets*. ... So, in fact, we have a colouring of the non-loop edges of the*graph*of (A, ≤ 1 ) into two colours: blue*and*red, such that each coloured component induces a*poset**on*A. ...##
###
Page 5063 of Mathematical Reviews Vol. , Issue 88j
[page]

1988
*
Mathematical Reviews
*

Let

*k*be a field*and*R a*homogeneous**k*-algebra. Let Q C Rj, the second direct summand of R, be a finite*poset*. ... The author characterizes all trees (rank*one*connected*posets*without cycles) which are integral over an infinite field*k*. Michael E. ...##
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Page 6103 of Mathematical Reviews Vol. , Issue 2001I
[page]

2001
*
Mathematical Reviews
*

Earlier work of Bezrukov

*on*Macaulay*posets*led to considering products of spider*posets*Q(*k*,1) consisting of*k*+ 1 incomparable chains of size /*and*an extra element that lies above them all. ... West (1-IL; Urbana, IL) 2001i:06005 06A12 20B27 Droste, Manfred (D-TUD-AL; Dresden); Kuske, Dietrich (D-TUD-AL; Dresden) ; Truss, John*K*. (4-LEED-PMS; Leeds)*On**homogeneous*semilattices*and*their automorphism ...##
###
Duality of graded graphs through operads
[article]

2020
*
arXiv
*
pre-print

This pair of

arXiv:2004.12230v1
fatcat:73olaixwhzdqtjdl4hqdrij3jm
*graphs*is dual for a new notion of duality called ϕ-diagonal duality, similar to the*ones*introduced by Fomin. ... We also provide a general way to build pairs of graded*graphs*from operads, wherein underlying*posets*are analogs to the Young lattice. ... Due to the fact that G is combinatorial*and**K*G is a graded space decomposing as*K*G = ∈N*K*G( ) (1.1.7) with finite dimensional*homogeneous*components*K*G( ) , 0, the space*K*G can be identified with ...##
###
Ramsey classes of topological and metric spaces

2006
*
Annals of Pure and Applied Logic
*

This relates the classification programme for countable

doi:10.1016/j.apal.2005.07.004
fatcat:zbnia2gxabanvbq75rngmhlyhi
*homogeneous*structures (of Lachlan*and*Cherlin) to the proof techniques of the structural Ramsey theory. ... Here we consider the classes of topological*and*metric spaces which recently were studied in the context of extremally amenable groups*and*of the Urysohn space. ... Petersburg meeting,*and*Professor Anatoly Vershik in particular, for a very nice meeting*and*hospitality. I also thank Andreas Blass*and*Aleš Pultr for several remarks related to this paper. ...##
###
Page 1735 of Mathematical Reviews Vol. , Issue 87d
[page]

1987
*
Mathematical Reviews
*

The proof is based

*on*the fact that the existence of a 6-central element in every height-*one**poset*is equivalent to a certain prop- erty of bipartite*graphs*. ... Finally, as a consequence of these detailed studies, the author is able to show that, although*k*-transitivity is indeed weaker than*k*-*homogeneity*for finite*k*, w-transitivity*and*w-*homogeneity*are equivalent ...##
###
A new operation on partially ordered sets
[article]

2012
*
arXiv
*
pre-print

As an corollary of our results

arXiv:1208.3504v3
fatcat:mekcigdlojgztl2j3guos4yuem
*on*rotations of finite*posets*, we obtain that the group of rotating permutations of the random*poset*is the automorphism group of a*homogeneous*structure in a finite language ... In this paper we introduce rotations for finite*posets*, which can be seen as the*poset*counterpart of Seidel-switch for finite*graphs*. ... Moreover, if A is finite,*and*a 1 , . . . , a*k*∈ A enumerate A in such a way that a j < a i for all 1 ≤ i < j ≤*k*, then (3) C A = C a*k*C a*k*−1 · · · C a 2 C a 1 . Proof. ...##
###
Page 666 of Mathematical Reviews Vol. , Issue 89B
[page]

1989
*
Mathematical Reviews
*

A rooted

*graph*(G, x) is a rooted subgraph of a rooted*graph*(F, y) if there exists a*one*-to-*one*function /:V(G) — V(F) that satisfies (1) [u,v] € E(G) >[f(u), f(v)] € E(F),*and*(2) f(x) = y. ... The rotation number /(G, x) of the p-vertex rooted*graph*(G, x) is defined to be the smallest number q(F) of edges in a p-vertex*graph*F such that (G, x) is a*homogeneous*rooted subgraph of F. ...##
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A new operation on partially ordered sets

2013
*
Journal of combinatorial theory. Series A
*

As an corollary of our results

doi:10.1016/j.jcta.2013.04.003
fatcat:e245iblv3fbnfcmv72h6voz4gq
*on*rotations of finite*posets*, we obtain that the group of rotating permutations of the random*poset*is the automorphism group of a*homogeneous*structure in a finite language ... In this paper we introduce rotations for finite*posets*, which can be seen as the*poset*counterpart of Seidel-switch for finite*graphs*. ... Moreover, if A is finite,*and*a 1 , . . . , a*k*∈ A enumerate A in such a way that a j < a i for all 1 ≤ i < j ≤*k*, then (3) C A = C a*k*C a*k*−1 · · · C a 2 C a 1 . Proof. ...
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