Lajos Soukup

Sticks and clubs

Abstract We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martins axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω 1 together with ¬CH and Martins axiom for countable p.o.-sets.

More on countably compact, locally countable spaces

Following [5], aT3 spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>ω+1+┐G(ωω+1); (iii) the “Chang conjecture” (ωω+1),→(ω1,ω) implies ┐S(κ) for allκ≧k≧ωω; (iv) ifP addsω1 dominating reals toV iteratively then, in , we haveG(λω) for allλ.

Indestructible properties of S- and L-spaces

Abstract Building on a method of Abraham and Todorcevic we prove a preservation theorem on certain properties under c.c.c. forcings. Applying this result we show that (1) an uncountable, first countable, 0-dimensional space containing only countable and co-countable open subspaces, and (2) S- and L-groups can exist under Martins Axiom.

On the weak Freese-Nation property of complete Boolean algebras

Abstract The following results are proved: (a) In a model obtained by adding ℵ 2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof ([μ] ℵ 0 ,⊆)=μ + hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of 〈 P (ω),⊆〉 is equivalent to the weak Freese-Nation property of any of C (κ) or R (κ) for uncountable κ. (d) Modulo the consistency of (ℵ ω+1 ,ℵ ω )↠(ℵ 1 ,ℵ 0 ) , it is consistent with GCH that C (ℵ ω ) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P (ω), ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.

Approximate Counting of Graphical Realizations

In 1999 Kannan, Tetali and Vempala proposed a MCMC method to uniformly sample all possible realizations of a given graphical degree sequence and conjectured its rapidly mixing nature. Recently their conjecture was proved affirmative for regular graphs (by Cooper, Dyer and Greenhill, 2007), for regular directed graphs (by Greenhill, 2011) and for half-regular bipartite graphs (by Miklós, Erdős and Soukup, 2013). Several heuristics on counting the number of possible realizations exist (via sampling processes), and while they work well in practice, so far no approximation guarantees exist for such an approach. This paper is the first to develop a method for counting realizations with provable approximation guarantee. In fact, we solve a slightly more general problem; besides the graphical degree sequence a small set of forbidden edges is also given. We show that for the general problem (which contains the Greenhill problem and the Miklós, Erdős and Soukup problem as special cases) the derived MCMC process is rapidly mixing. Further, we show that this new problem is self-reducible therefore it provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for counting of all realizations.With the current burst of network theory research (especially in connection with social and biological networks) there is a renewed interest on realizations of given degree sequences and uniform sampling of those realizations. In this paper we propose a new degree sequence problem: we want to find graphical realizations of a given degree sequence on labeled vertices, where certain would-be edges are forbidden. Then we want to sample uniformly all possible realizations. In this paper, as a first step, we solve this restricted degree sequence (or RDS for short) problem if the forbidden edges form a bipartite graph, which consist of the union of a (not necessarily maximal) 1-factor and a (possible empty) star. Then we show how one can sample the space of all realizations of those RDSs uniformly if the degree sequence describes half-regular bipartite graphs. Our result contains, as special cases, the wellknown result of Kannan, Tetali and Vempala on sampling regular bipartite graphs and a recent result of Greenhill on sampling regular directed graphs (so it also provides new proofs of them). The RDS problem with one forbidden 1-factor and with one star is self-reducible, therefore our fully polynomial almost uniform sampler (a.k.a. FPAUS) on the space of all realizations also provides a fully polynomial randomized approximation scheme (a.k.a. FPRAS) for approximate counting of all realizations.

How To Split Antichains In Infinite Posets

A maximal antichain A of poset P splits if and only if there is a set B ⊂ A such that for each p ∈ P either b ≤ p for some b ∈ B or p ≤ c for some c ∈ A\B. The poset P is cut-free if and only if there are no x < y < z in P such that [x,z]P = [x,y]P ∪ [y,z]P . By [1] every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see [2])) we prove here that if a maximal antichain in a cut-free poset “resembles” to a finite set then it splits. We also show that a version of this theorem is just equivalent to Axiom of Choice.We also investigate possible strengthening of the statements that “A does not split” and we could find a maximal strengthening.

A tall space with a small bottom

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if • <• = • then there is such a space of height • + with onlymany isolated points. This implies that there is a locally compact scattered space of height !2 with !1 isolated points in ZFC, solving an old problem of the flrst author.

How to force a countably tight, initially ω1-compact and noncompact space?☆

Abstract Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional, Frechet-Urysohn, initially ω 1 -compact and noncompact space X of size ω 2 having the following property: for every open (or closed) set A in X we have ¦ A ¦ ⩽ ω 1 or ¦ X β A ¦ ⩽ ω 1 .

Hereditary normality of γN-spaces

A γN-space is a locally compact Hausdorff space with a countable dense set of isolated points, and the rest of the space homeomorphic to ω1. We show that under the Open Coloring Axiom (OCA) no γN-space is hereditarily normal. This is the key to showing that some sweeping statements are consistent with (and independent of) the usual axioms of set theory, including: 1. (1) Every countably compact, hereditarily normal space is sequentially compact. 2. (2) Every separable, hereditarily normal, countably compact space is compact and Frechet-Urysohn. 3. (3) The arbitrary product of countably compact, hereditarily normal spaces is countably compact. Not all of these conclusions follow just from MA + ¬ CH: a forcing construction is given of a model of MA + c = κ where κ is any cardinal ⩾ ℵ2 satisfying κ = 2

Elementary submodels in infinite combinatorics

The use of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williamss theorem on cycle decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.