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

Norbert Sauer
2001 Combinatorica
Assume that 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]

Natasha Dobrinen
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)

Reinhard Diestel, Imre Leader, Alex Scott, Stéphan Thomassé
2018 Transactions of the American Mathematical Society
These 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

Jaroslav Nes̆etr̆il, Vojtĕch Rödl
1985 Discrete Mathematics
We present a very simple proof 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

András Gyárfás
1998 Journal of combinatorial theory. Series A
We consider finite undirected simple 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]

Robert Cowen
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]

Vikram Kamat, Neeldhara Misra
2013 arXiv   pre-print
We consider 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

P.E. Haxell, Y. Kohayakawa
1996 Journal of combinatorial theory. Series B (Print)
Any r-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 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

Zbigniew Lonc
1997 Discrete Mathematics
We show that for any positive integer c 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

P Erdős, A Gyárfás, L Pyber
1991 Journal of combinatorial theory. Series B (Print)
If 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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