Abstract. We will present a collection of guessing principles which have a similar relationship to ♦ as cardinal invariants of the continuum have to CH. The purpose is to provide a means for systematically analyzing ♦ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of CH and ♦ in models such as those of Laver, Miller, and Sacks.

Abstract. We study combinatorial properties of the partial order (Dense(Q), ⊆).To do that we introduce cardinal invariants p Q , t Q , h Q , s Q , r Q , i Q describing properties of Dense(Q). These invariants satisfyWe compare them with their analogues in the well studied Boolean algebra P(ω)/fin. We show that p Q = p, t Q = t and i Q = i, whereas h Q > h and r Q > r are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of Q. In particular, we show that non(M) = min{|D| :We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah. 0. Introduction. The aim of this paper is to point out the similarities and differences between the structure of P(ω)/fin and the structure of the collection of dense subsets of the rationals. Such research was suggested by A. Blass in [Bl] and initiated by J. Cichoń in [Ci]. The basic object studied here is the set Dense(Q) = {D ⊆ Q : D is dense} ordered by inclusion, in comparison with the structure ([ω] ω , ⊆). Neither one of them is a separative partial order. The separative quotient of ([ω] ω , ⊆) augmented with the least element 0 is the well known Boolean algebra P(ω)/fin. The separative quotient of (Dense(Q), ⊆) with added least element is not a Boolean algebra, but just a lattice, with two dense sets being in the same equivalence class if and only if their symmetric difference is a 2000 Mathematics Subject Classification: 03E17, 03E35, 06E15. Key words and phrases: rational numbers, nowhere dense ideal, distributivity of Boolean algebras, cardinal invariants of the continuum.

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Abstract. We study the cardinal invariants of analytic P-ideals, concentrating on the ideal Z of asymptotic density zero. Among other results we prove min{b, cov (N)} ≤ cov * (Z) ≤ max{b, non(N)}.

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting that if we color edges of the complete graph we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák and Shelah [5] are answered by showing that assuming the consistency of suitable large cardinals the following are relatively consistent with ZFC:Building on a work of Lambie-Hanson [14], we also show thatTo prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

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