Partitions of large Rado graphs

Let κ be a cardinal which is measurable after generically adding ℶκ+ω many Cohen subsets to κ and let G = (κ, E) be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value r + m such that the set [κ] m can be partitioned into classes ˙ Ci: i < r + ¸ m such that for any coloring of any of the classes Ci in fewer than κ colors, there is a copy G ∗ of G in G such that [G ∗ ] m ∩ Ci is monochromatic. It follows that G → (G) m <κ/r + m that is, for any coloring of [G] m with fewer than κ colors there is a copy G ′ of G such that [G ′ ] m has at most r + m colors. On the other hand, we show that there are colorings of G such that if G ′ is any copy of G then Ci ∩ [G ′ ] m ̸ = ∅ for all i < r + m, and hence G ↛ [G] m r + m We characterize r + m as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, r + 2 and for m> 2 we have r + m> rm where rm is the corresponding number of types for the countable Rado graph. = 2

Strongly intersecting integer partitions

We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe

Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

Packing Steiner Trees

Let $T$ be a distinguished subset of vertices in a graph $G$. A $T$-\emph{Steiner tree} is a subgraph of $G$ that is a tree and that spans $T$. Kriesell conjectured that $G$ contains $k$ pairwise edge-disjoint $T$-Steiner trees provided that every edge-cut of $G$ that separates $T$ has size $\ge 2k$. When $T=V(G)$ a $T$-Steiner tree is a spanning tree and the conjecture is a consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved that Kriesell's conjecture holds when $2k$ is replaced by $24k$, and recently West and Wu have lowered this value to $6.5k$. Our main result makes a further improvement to $5k+4$.Comment: 38 pages, 4 figure

Is Ramsey's theorem omega-automatic?

We study the existence of infinite cliques in omega-automatic (hyper-)graphs. It turns out that the situation is much nicer than in general uncountable graphs, but not as nice as for automatic graphs. More specifically, we show that every uncountable omega-automatic graph contains an uncountable co-context-free clique or anticlique, but not necessarily a context-free (let alone regular) clique or anticlique. We also show that uncountable omega-automatic ternary hypergraphs need not have uncountable cliques or anticliques at all