Jean A. Larson

Pairwise balanced designs whose line sizes do not divide six

Abstract A proper PBD(L, v) is a pairwise balanced design (with λ = 1) on v points whose line sizes are less than v and do not divide six. The existence or nonexistence of a proper PBD(L, v) is determined for all v different from thirty.

A quest for certain linear spaces on thirty points

Abstract We investigate the structure of linear spaces on 30 points which have no lines of sizes 2, 3, 6, 30.

Some graphs with chromatic number three

Abstract A question of P. Erdos is solved by showing that certain graphs have chromatic number at most three. The proof proceeds by showing a conjecture of Erdos and Bollobas holds, namely, that under certain circumstances, a graph which contains an odd circuit must contain an odd circuit with diagonal.

A counter-example in the partition calculus for an uncountable ordinal

The main theorem of the paper is a counter-example in the partition calculus introduced by P. Erdös and R. Rado: Ifκ is a regular cardinal and α=↛(α,3)2, then α ↛ (α,3)2. The proof is combinatorial. Other counter-examples are produced from this one through the pinning relation which was introduced by E. Specker.

The quasi order of graphs on an ordinal

For @a an ordinal, a graph with vertex set @a may be represented by its characteristic function, f:[@a]^2->2, where f({@c,@d})=1 if and only if the pair {@c,@d} is joined in the graph. We call these functions @a-colorings. We introduce a quasi order on the @a-colorings (graphs) by setting f@?g if and only if there is an order-preserving mapping t:@a->@a such that f({@c,@d})=g({t(@c),t(@d)}) for all {@c,@d}@?[@a]^2. An @a-coloring f is an atom if g@?f implies f@?g. We show that for @a=@w^@w below every coloring there is an atom and there are continuum many atoms. For @a<@w^@w below every coloring there is an atom and there are finitely many atoms.

On a problem of Erdős and Rado

AbstractWe give some improved estimates for the digraph Ramsey numbersr(Kn*,Lm), the smallest numberp such that any digraph of orderp either has an independent set ofn vertices or contains a transitive tournament of orderm.By results of Baumgartner and of Erdős and Rado, this is equivalent to the following infinite partition problem: for an infinite cardinal κ and positive integersn andm, find the smallest numberp such that

The number of one-generated diagonal-free cylindric set algebras of finite dimension greater than two

\kappa \cdot p \to (\kappa \cdot n,m)^2

Matchings from a set below to a set above

that is, find the smallest numberp so that any graph whose vertices are well ordered where order type κ·p either has an independent subset of order type κ·n or a complete subgraph of sizem.