Fred Galvin

The list chromatic index of a bipartite multigraph

Abstract For a bipartite multigraph, the list chromatic index is equal to the chromatic index (which is, of course, the same as the maximum degree). This generalizes Janssen′s result on complete bipartite graphs Km, n with m ≠ n; in the case of Kn, n, it answers a question of Dinitz. (The list chromatic index of a multigraph is the least number n for which the edges can be colored so that adjacent edges get different colors, the color of each edge being chosen from an arbitrarily prescribed list of n different colors associated with that edge.)

γ-sets and other singular sets of real numbers

Abstract A family of J of open subsets of the real line is called an ω-cover of a set X iff every finite subset of X is contained in an element of J . A set of reals X is a γ-set iff for every ω-cover J of X there exists 〈D n : n J ω such that X⊆ ∪ n ∩ m > n D m . In this paper we show that assuming Martins axiom there is a γ-set X of cardinality the continuum.

Stationary strategies in topological games

Abstract We prove general theorems on the existence of stationery strategies (i.e., strategies depending only on the opponents last move) in certain infinite positional games of perfect information and we derive some consequences for various topological games.

Inequalities for Cardinal Powers

Silver [7] recently proved that, if GCH holds below , then it holds at 8 [Lo2lo]wr for some p < 03. (Here (wo, is an ordinal power; in the rest of this paper, only cardinal exponentiation is used.) We thank Prikry and Silver for communicating their results to us.

On the density of λ-box products

An apparatus removing air from pressurized gas, including a pressure housing removably securable to a connection cap, a removable filter housing in the housing with moisture removing material in the filter housing, and an external indicator indicating that the apparatus is not removing at least some moisture. The cap includes an air flow exit for air which has passed through the filter housing. The indicator is on the cap and includes visible material which changes color when moisture is present. An aspirator tube diverts some air flow from the cap air flow exit into the indicator visible color changing material. The filter housing is transparent or translucent and includes oil removing material and a sintered disk before the moisture removing material.

Some counterexamples in the partition calculus

We show that the pairs (2-element subsets; edges of the complete graph) of a set of cardinality ℵ 1 can be colored with 4 colors so that every uncountable subset contains pairs of every color, and that the pairs of real numbers can be colored with ℵ 0 colors so that every set of reals of cardinality 2 ℵ0 contains pairs of every color. These results are counterexamples to certain transfinite analogs of Ramseys theorem. Results of this kind were obtained previously by Sierpinski and by Erdos, Hajnal, and Rado. The Erdos-Hajnal-Rado result is much stronger than ours, but they used the continuum hypothesis and we do not. As by-products, we get an uncountable tournament with no uncountable transitive subtournament, and an uncountable partially ordered set such that every uncountable subset contains an infinite antichain and a chain isomorphic to the rationals. The tournament was pointed out to us by R. Laver, and is included with his permission.

A Proof of Dilworth's Chain Decomposition Theorem

Let P be a finite partially ordered set: a chain (antichain) in P is a set of pairwise comparable (incomparable) elements; the width of P is the maximum cardinality of an antichain in P. According to a celebrated theorem of Dilworth [2], the width of P is also equal to the minimum number of chains needed to cover P. The wider combinatorial significance of Dilworths theorem, especially as regards matching theory, is discussed by Bogart, Greene, and Kung [1], Mirsky [3], and Reichmeider [5]. Bogart, Greene, and Kung survey various proofs of Dilworths theorem; the proofs of Perles [4] and Tverberg [6] are especially simple and elegant. The proof given here seems to me to be as simple as any. This proof is probably well-known folklore; still, as far as I know, it has never appeared in print.

A Ramsey-type theorem for traceable graphs☆

Abstract A graph G is traceable if there is a path passing through all the vertices of G . It is proved that every infinite traceable graph either contains arbitrarily large finite chordless paths, or contains a subgraph isomorphic to graph A , illustrated in the text. A corollary is that every finitely generated infinite lattice of length 3 contains arbitrarily large finite fences . It is also proved that every infinite traceable graph containing no chordless four-point path contains a subgraph isomorphic to K ω,ω . The versions of these results for finite graphs are discussed.

Another note on cliques and independent sets

In this paper, we prove that any graph G with maximum degree

Some Ramsey-type theorems

\Delta(G)\geq(11+\sqrt{49-24_{\chi}(\Sigma)})/2