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Another Look at the Erdős-Hajnal-Pósa Results on Partitioning Edges of the Rado Graph

2001
*
Combinatorica
*

Assume that

doi:10.1007/s004930100026
fatcat:a457h5nmynfp5ahntfx2swzkn4
*the*set R*of*vertices*of**the**Rado**graph*is ordered under < into a total order*of*type ω. Let U be*the*set*of*up*edges*and D be*the*set*of*down*edges**of*R under*the*order <. ... This defining property*of**the**Rado**graph*is called*the*mapping extension property.*The**Rado**graph*is one*of**the*prime examples*of*a homogeneous structure. [2] . Let H be a countable*graph*. ...##
###
A List of Problems on the Reverse Mathematics of Ramsey Theory on the Rado Graph and on Infinite, Finitely Branching Trees
[article]

2018
*
arXiv
*
pre-print

*The*intent is to enlist

*the*help

*of*those working in Reverse Mathematics to take on such problems, and

*the*myriad

*of*related questions one can infer from them. ... This list presents problems in

*the*Reverse Mathematics

*of*infinitary Ramsey theory which I find interesting but do not personally have

*the*techniques to solve. ... (11) Does

*the*rainbow Ramsey theorem for

*the*

*Rado*

*graph*have

*the*same reverse math strength as

*the*canonical

*partition*theorem

*of*Sauer? ...

##
###
:{unav)

2018
*
Transactions of the American Mathematical Society
*

These

doi:10.1090/s0002-9947-06-04086-4
fatcat:n42c7wefhffxlfyy34dbck7asm
*graphs*are*the*edgeless*graph*,*the*random tournament,*the*transitive tournaments*of*order type ω α , and two orientations*of**the**Rado**graph*:*the*random oriented*graph*, and a newly found random acyclic ...*The*inverse*of*G is*the*oriented*graph*obtained from G by reversing*the*directions*of*all its*edges*. ... This completes*the*proof*of*Lemma 5.1, and*of*Theorem 2.3. § Let us close by mentioning that we do not know whether or not*the**Rado**graph*R is '*edge*-indivisible', in*the*sense that whenever we*partition*...##
###
Two remarks on Ramsey's theorem

1985
*
Discrete Mathematics
*

We present a very simple proof

doi:10.1016/0012-365x(85)90118-9
fatcat:6jdfblykinfafindukffgjhgfq
*of**the*fact (due to P. Erd~s and R.*Rado*) that Ram~ey's theorem doesn't hold for*partitions**of*infinite subsets. ...*Rado*) and to*the*statement VG ::IHH---, (G)~, where both G and H are finite*graphs*(due to W. ... ~~s*of*~ite subsets Recall that, given cardinals a, /3, % 6,*The*Erd6s-*Rado**partition*arrow --~ (a)~ has*the*following meaning: for every*partition*c : [ . ...##
###
Page 1730 of Mathematical Reviews Vol. , Issue 87d
[page]

1987
*
Mathematical Reviews
*

One

*of**the*most frequently formulated problems is that*of*optimal*partitioning**of*nodes*of*a given hypergraph H in order to maximize*the*sum*of*interclass connections (*edges**of*H). ... A*graph*in which each*edge*is assigned*the*value +1 or —1 is called a signed*graph*. A balanced signed*graph*has each cycle with an COMBINATORICS 1730 even number*of*negative*edges*. ...##
###
Page 4450 of Mathematical Reviews Vol. , Issue 84k
[page]

1984
*
Mathematical Reviews
*

“A maximal-clique

*partition**of*a*graph*G is a way*of*covering G with maximal complete subgraphs such that every*edge*belongs to exactly one*of**the*subgraphs. ... Authors’ summaries: “In part IV we present linear time algorithms for computing*the*minimum number*of*complete subgraphs needed to cover or*partition**the**edges**of*any simple*graph*G with maximal degree ...##
###
Generalized Split Graphs and Ramsey Numbers

1998
*
Journal of combinatorial theory. Series A
*

We consider finite undirected simple

doi:10.1006/jcta.1997.2833
fatcat:33kqbzngareyxd4kz7vzkglntq
*graphs*G=(V, E), where V, E are*the*vertex set and*edge*set*of*G, respectively.*The*numbers |V|, |E| are called*the*order and*the*size*of**the**graph*G. ... A*graph*G is called a ( p, q)-split*graph*if its vertex set can be*partitioned*into A, B so that*the*order*of**the*largest independent set in A is at most p and*the*order*of**the*largest complete subgraph ...*The*sets X i and Y i form hypergraphs with m*edges*and*the*rank*of*both hypergraphs is at most r. Set g( p, q)=F(F(r)) where F is*the*Erdo s*Rado*function from Theorem A. ...##
###
Integrated Neighborhood Colorings of Graphs
[article]

2022
*
arXiv
*
pre-print

*The*idea that those different from you are "unfriendly" is captured in

*the*definition

*of*unfriendly 2-colorings in

*graph*theory in a paper by Aharoni, Milner and Prikry, where they prove that every finite ...

*graph*has an unfriendly coloring. ... Let G be a finite

*graph*with m

*edges*.

*The*

*the*vertices

*of*G can be

*partitioned*into k sets: V 1 , ... , V k with at least ((k-1)/k )m

*edges*with vertices in different sets

*of*

*the*

*partition*. Proof. ...

##
###
Page 3609 of Mathematical Reviews Vol. , Issue 2001F
[page]

2001
*
Mathematical Reviews
*

and/or

*edges**of*G are subsets*of**the*sets*of*vertices and/or*edges**of*H). ...*The*second half*of**the*paper deals with*the**partition*lattice*of*finite sets. Three definitions for*the*intersection*of**partitions*are given. ...##
###
An Erdős--Ko--Rado theorem for matchings in the complete graph
[article]

2013
*
arXiv
*
pre-print

We consider

arXiv:1303.4061v1
fatcat:ddumxps7nbeg3c5ftd665flaca
*the*following higher-order analog*of**the*Erdős--Ko--*Rado*theorem. For positive integers r and n with r<= n, let M^r_n be*the*family*of*all matchings*of*size r in*the*complete*graph*K_2n. ... For any*edge*e in E(K_2n),*the*family M^r_n(e), which consists*of*all sets in M^r_n containing e, is called*the*star centered at e. ... Baranyai*Partitions*A Baranyai*partition**of**the*complete*graph*K 2n is a*partitioning**of*its*edge*-set E into (2n − 1) perfect matchings. ...##
###
Partitioning by Monochromatic Trees

1996
*
Journal of combinatorial theory. Series B (Print)
*

Any r-

doi:10.1006/jctb.1996.0065
fatcat:7ueyqnojcfbe7otdjjpejiqde4
*edge*-coloured n-vertex complete*graph*K n contains at most r monochromatic trees, all*of*different colours, whose vertex sets*partition**the*vertex set*of*K n , provided n 3r 4 r! ... This comes close to proving, for large n, a conjecture*of*Erdo s, Gya rfa s, and Pyber, which states that r&1 trees suffice for all n. ... INTRODUCTION*The*tree*partition*number*of*r-*edge*-coloured complete*graphs*is defined to be*the*minimum k such that whenever*the**edges**of*a complete*graph*K n are coloured with r colours,*the*vertices*of*...##
###
An Asymmetric Random Rado Theorem: 1-statement
[article]

2019
*
arXiv
*
pre-print

A classical result by

arXiv:1906.05614v2
fatcat:emmolfcqrjccnfckks35taali4
*Rado*characterises*the*so-called*partition*-regular matrices A, i.e. those matrices A for which any finite colouring*of**the*positive integers yields a monochromatic solution to*the*... ,A_r are predetermined*partition*-regular matrices pre-assigned to*the*colours involved. We prove a 1-statement for*the*asymmetric random*Rado*property. ...*the**partition*ξ), and we refer to such*edges*as ξ-*partite*. ...##
###
Page 3807 of Mathematical Reviews Vol. , Issue 85i
[page]

1985
*
Mathematical Reviews
*

From this follow a number

*of*known and new canonical Ramsey theorems, such as*the*Erdés-*Rado*canonization theorem and Taylor’s canonical*Rado*-Folk- man-Sanders finite unions theorem. R. L. ... (5 + =a) where |e;(Q,,)| denotes*the*number*of**edges**of*Q, which are coloured with*the*ith colour, i = 1,2. Theorem 3 is a variation*of*extremal*graph*theorems such as Turdn’s theorem. ...##
###
Delta-system decompositions of graphs

1997
*
Discrete Mathematics
*

We show that for any positive integer c

doi:10.1016/s0012-365x(96)00054-4
fatcat:yacwfgmlzvfq5dxrgzk2lm4egm
*the*problem whether*the**edge*-set*of*a*graph*can be*partitioned*into subsets inducing*graphs*isomorphic to either a c-*edge*star or a c-*edge*matching is polynomial ... This result suggests existence*of*theorems well-characterizing*graphs*admitting such*partitions*. 0012-365X/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved PH S0012-365X(96)00054-4 ... We say that a*graph*G has a decomposition into*graphs*G1 ..... Gt if*the*sets*of**edges**of*G1 ..... G t form a*partition**of**the*set*of**edges**of*G. ...##
###
Vertex coverings by monochromatic cycles and trees

1991
*
Journal of combinatorial theory. Series B (Print)
*

If

doi:10.1016/0095-8956(91)90007-7
fatcat:dkyocj5f3jfvtdkqngr6e72nay
*the**edges**of*a finite complete*graph*K are colored with r colors then*the*vertex set*of*K can be covered by at most cr2 log r vertex disjoint monochromatic cycles. ... ACKNOWLEDGMENTS*The*authors are grateful to referees whose remarks led to reorganizing*the*original version*of*this paper. ... This is a result*of**Rado*[15] .*The*rest*of*this section is devoted to tree cover and tree*partition*numbers. ...
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