History of infinity in set theory
Saeed Seyed Agha Banihashemi, Dept. of Math and IT School of International Relation, Ambassador of I.R.Iran in Hungary ,Stefania ut 97 1143 Budapest
Coauthors: no

In this article I am to talk about the history of infinity. As we know history of mathematical symbol can help us in teaching, applications and research to do our job better. After talking about how infinity invented, I will talk about on infinite set theory.

Universal minimal flows for groups of automorphisms of homogeneous relational structures
Dana Bartosova, Charles University in Prague, University of Toronto

Universal minimal flows are objects of topological dynamics of major interest. We use ultrafilter flows and Ramsey theorems for classes of finite structures to give a simple computation of universal minimal flows of dense subgroups of groups of automorphisms of homogeneous relational structures, it means closed subgroups of permutation groups. In particular, we show that the universal minimal flows for the group of homeomorphisms of w^* and the group of trivial homeomorphisms of w^* are the same in the realm of ZFC. This work unifies and generalizes proofs of Glasner and Weiss, Glasner and Gutman and Pestov and generalizes results of Kechris, Pestov and Todorcevic on countable structures and Balcar and Franek on discrete groups.

References

B. Balcar, F. Franek Structural properties of universal minimal dynamical systems for discrete semigroups, Tran. Amer. Math. Soc. 349 (1997) 1697-1724.

E. Glasner, B. Weiss The universal minimal system for the group of homeomorphisms of the Cantor set, Fund. Math. 176 (2003), 277-289.

E. Glasner, Y. Gutman The universal minimal space for groups of homeomorphisms of h-homogeneous spaces, preprint.

A.S. Kechris, V.G. Pestov, S. Todorcevi\'c Fraiïssé limits, Ramsey theory, and topological dynamics of automorphism groups, GAFA Geom. funct. anal. 15 (2005) 106-189.

V. Pestov Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon, American Mathematical Society, Providence, RI (2006), viii+192.

On Hajnal's Triangle Free Game
Csaba Biro, University of Louisville
Coauthors: Paul Horn, D. Jacob Wildstrom

Hajnal's triangle free game starts on the empty graph on n vertices. Two players take turns in adding edges to the graph while keeping the graph triangle free. In Hajnal's original game, the player who could not make a valid move lost the game. Füredi, Reimer, and Seress proposed a variation, in which the goal of one of the players is to prolong the game, and the goal of the other player is to finish the game as soon as possible. They proved that the game will last at least on the order of nlogn steps. On the other hand, an unpublished result by Erdös states that the game will last at most n²/5 steps. In this paper we make a slight improvement on Erdös's upper bound and we discuss related problems.

Measures on Bell space
Piotr Borodulin-Nadzieja, Instytut Matematyczny, Wrocław University, Poland
Coauthors: Mirna D?amonja

We show that there is a compact space which is separable, which does not have a countable pi-base and which carries only separable measures. This is a modification of an example due to Murray Bell [1]. Some consequences of the existence of such a space will be presented.

[1] M. Bell, G_k subspaces of hyadic spaces, PAMS 104/2 (1988)

Cardinal functions F_q(X) and t_q(X) for H-closed spaces
Andrei Catalioto, University of Messina (Italy)
Coauthors: Filippo Cammaroto (University of Messina) - Jack Porter (University of Kansas)

In this article, starting with results and answering questions by Cammaroto and Kocinac in 1993, we prove that t_q(X) = t_q(X_s) = t(X_s) = F_q(X) = F_q(X_s) = F(X_s) for every H-closed and Urysohn space and we present some useful examples to study the full relations between cardinal functions t, t_q, F and F_q in the context of H-closed and Urysohn spaces.
Also, we construct an H-closed space H for which F_q(H) < t_q(H).

References:
[1]  A.V. Arhangel'skii, On bicompacta hereditarily satisfying Suslin's condition. Tightness and free sequences, Soviet Math. Dokl., 12 (1971), 1253-1257.
[2]  F. Cammaroto, A. Catalioto and J. Porter, Cardinal functions F_q(X) and t_q(X) for H-closed spaces (2011), preprint.
[3]  F. Cammaroto and Lj.D. Kocinac, On q-tightness, Facta Universitatis (Nis), Ser. Math. Inform. 8 (1993), 77-85.
[4]  F. Cammaroto and Lj.D. Kocinac, A note on q-tightness, Rend. Circ. Mat. Palermo, Ser II 42 (1993), 129-134.
[5]  I. Juhász, Cardinal Functions in Topology - Ten Years Later, Math. Centre Tracts 123, Amsterdam, 1980.
[6]  J.R. Porter and R.G. Woods, Extensions and absolutes of Hausdorff spaces, Springer-Verlag, 1988.

Note:  q = theta (greek letter).

On hypergraph cliques with chromatic number 3
D.D. Cherkashin, Saint-Petersburg State University
Coauthors: A.M. Raigorodskii

Canonization Theorems on Ramsey spaces and their application to the Tukey theory of ultrafilters
Natasha Dobrinen, University of Denver
Coauthors: Stevo Todorcevic, CNRS University of Paris 7

We present canonization theorems for countable colorings on fronts on new topological Ramsey spaces. These Ramsey spaces generalize the Ellentuck space and are built to correspond to some forcings of Laflamme which add weakly Ramsey ultrafilters and weaker variations. The canonization theorems start with the Erdos-Rado canonization theorem and build to obtain canonization theorems which are the analogs of the Erdos-Rado theorem and the Pudlak-Rodl theorem for these Ramsey spaces. These canonization theorems are then applied to characterize the Tukey types of ultrafilters below the Laflamme ultrafilters in terms of the Rudin-Keisler ordering.

Some cardinal invariants related to analytic quotients
Jana Flaskova, University of West Bohemia in Pilsen

We investigate cardinal invariants of the quotient algebra P(N)/W where W is the van der Waerden ideal - an ideal on the set of the natural numbers N which consists of the sets which do not contain arbitrarily long arithmetic progressions. We prove, in particular, that the pseudointersection number of P(N)/W is the same as the standard pseudointersection number defined for P(N)/fin, and almost disjointness number of P(N)/W is less or equal to the standard cardinal characteristics of the continuum with the same name.

Combinatorics and Large Cardinals
Sy-David Friedman, Kurt Goedel Research Center, University of Vienna

Andras Hajnal is renowned for his fundamental contributions to infinitary combinatorics, a field which lies at the heart of set theory. In this talk I will discuss some aspects of this field in the context of large cardinals. In the non-GCH setting I'll consider the behaviour of the generalised continuum function as well as cardinal characteristics associated to graph complexity, dominating families, the cofinality of the symmetric group and the tree property. In the GCH setting I'll discuss Jensen's Diamond, Square and Morass principles as well as forms of Gödel condensation. Additional topics concern the number of normal measures on a measurable cardinal and the impact of definability on combinatorial properties.

Classifying Ergodic Measure Preserving Transformations
Matthew Foreman, University of California, Irvine

In 1931 von Neumann proposed the project of classifying ergodic measure preserving transformations. This talk will describe how the tools of descriptive set theory can be used to show that such a classification is impossible, even for diffeomorphisms of the torus. The word "impossible" will be rigorously defined. The method of proof uses finite combinatorial arguments (randomization techniques) to build approximations of diffeomorphisms, thus illustrating another connection between the finitary techniques and set theory.

Quasi-selective and weakly Ramsey ultrafilters
Marco Forti, Dipart. Matematica Applicata "U. Dini", Universita` di Pisa

Recall that an ultrafilter U on N is selective iff every finite colouring of [N]² has a homogeneous set U in U iff every function is nondecreasing on some U in U. Natural weakenings of these properties led to the (inequivalent) notions of weakly Ramsey and of quasi-selective ultrafilter:
- U is weakly Ramsey (shortly WR) if for every finite colouring of [N]² there is U in U s.t. [U]² has only two colours (see [1]).

- U is f-quasi-selective (shortly f-QS) if every g ≤ f is nondecreasing on some set U in U. U is quasi-selective (shortly QS) if it is id-QS (see [2]).

All WR and all f-QS ultrafilters are P-points, so the following results are nontrivial only when such ultrafilters exist. However mild hypotheses, like CH or MA, suffice in making all these classes rich and distinct (see [1, 2]).

Theorem 1. Let U be a nonprincipal ultrafilter.
1. If U is f-QS, then there exists U in U s.t. x+f(x) < y for all x < y in U.
2. If g is increasing modulo U, then U is (f○g)-QS if and only if gU is f-QS.
3. If U is f-QS, then fU is QS, and U is selective if and only if it is rapid.

Let U be nonselective, let I_n=[a_n, b_n] be any interval partition witnessing the non-selectivity of U, and for x ∈ I_n put a^U(x)=|U∩[a_n, x)|, b^U(x)=|U∩ [x, b_n]|, c^U(x)=|U∩[a_n, b_n]|, e^U(x)=|U∩[0, x]|, and S^U={f |  f 1-1 on U}

Let A^U, B^U, C^U, E^U, and S^U be the cuts of the ultrapower N^N_U whose right parts are generated by {a^U | U ∈ U }, {b^U | U ∈ U }, {c^U | U ∈ U }, {e^U | U ∈ U}, and {S^U | U ∈ U } respectively.

Let F_U be the cut of the ultrapower N^N_U whose left part is generated by F_U={f | U f-QS}. Then, independently of the chosen interval partition,

Theorem 2. Let U be a nonprincipal nonselective ultrafilter.
1. If U is f-QS, then

N < FU ≤ BU ≤ max {AU, BU}=CU, and AU ≤ EU.

2. If U is WR, then

N ≤ min {AU, BU} = SU ≤ BU = FU ≤ max {AU, BU} = CU   and  AU = EU.

Moreover U is rapid if and only if N = S^U, and then all cuts coincide with N.

In particular a WR ultrafilter U is QS if and only if the identity is less than the cut C^U. More generally, we obtain a complete specification of the "quasi-selectivity" properties of WR ultrafilters:

Corollary Let U be a nonprincipal nonselective WR ultrafilter. Then
(i) U is f-QS for some f iff N ≠ C^U, or equivalently iff  U is not rapid;
(ii) U is f-QS for some f increasing modulo U if and only if A^U ≠ C^U;
(iii)  U is isomorphic to a QS ultrafilter if and only if  A^U ≠ B^U.

[1] A. BLASS, Ultrafilter mappings and their Dedekind cuts, Trans. Amer. Math Soc., 188 (1974), 327-340.

[2] A. BLASS, M. DI NASSO, M. FORTI, Quasi-selective ultrafilters and asymptotic numerosities, submitted.

Some Properties of Multidimensional Intersection Matrices
Armenak Gasparyan, PSI RAS, Pereslavl-Zalesskii

Given two finite sets X, Y and two families of their subsets F={U_i, U_i ⊆ X}, G={V_j, V_j ⊆ X}, the intersection matrix of F and G is a (m×n)-matrix A=(a_ij), where m=| F|, n=| G|, and a_ij=| U_i∩V_j|, i=1, ... m, j=1, ... n. More generally, let p be a positive integer, p ≥ 2, and let X₁, ... , X_p are given finite sets. For each r, r ∈ {1, ... , p}, we choose arbitrarily an n_r-element family F_r={U^(r)_ir, U^(r)_ir ⊆ X_r}, and for chosen families define a p-dimensional n₁×...×n_p-matrix A_p=∥a_{i1... ip}∥, i_r=1, ... , n_r, r=1, ... , p , that we call intersection matrix of F, F=(F₁, ... , F_p), in which a_{i1... \ip}=|X_i1∩...∩X_ip|.

The talk is devoted to the study of algebraic and combinatorial properties of matrices A_s, s=1, ... , p. Particularly, in case F₁=... = F_p or in more complex cases if F is dropped into groups of equal families, we are interested in positivity properties of higher order algebraic forms with A_s(F) as their coefficient matrices. We consider also some intersection matrices corresponding to another notions of intersection coming from different areas.

Clopen graphs and their finite induced subgraphs
Stefan Geschke, Hausdorff Center for Mathematics, Bonn

A graph G on a Polish space X is clopen if its edge relation is a subset of X²\{(x, x):x ∈ X} that is both closed and open.

From the perspective of descriptive set theory, clopen graphs are the least complex definable graphs. Cardinal invariants such as clique number and chromatic number are degenerate for clopen graphs in the sense that they are either countable or equal to 2^ℵ₀.

A cardinal invariant that is not degenerate is the cochromatic number, the least size of a family of cliques and independent sets that covers all vertices. Cochromatic numbers of clopen graphs on Polish spaces came up in the context of planar convexity. It turns out that the the cochromatic number of a clopen graph is largely determined by its finite induced subgraphs.

Definition 1. For a graph G let age(G) denote the class of all finite graphs isomorphic to induced subgraphs of G. If the set V(G) of vertices of G is a topological space, let the hereditary age hage(G) be the class of all finite graphs that have induced copies in all nonempty open subsets of V(G). G is self-similar if hage(G)=age(G).

Definition 2. Let G and F be graphs. A map f:V(G)→ V(F) is modular if for all v, w ∈ V(G) with f(v) ≠ f(w), {v, w} is an edge in G iff {f(v), f(w)} is an edge in F. A graph G whose set V(G) of vertices is a topological space is modular profinite if it is the limit of an inverse system of finite graphs with modular bonding maps.

Definition 3. For a class C of finite graphs let cl(C) denote the closure of C under isomorphism, induced subgraphs, and substitution. C is closed if C=cl(C). A class C is finitely generated if there is a finite set F of finite graphs such that C ⊆ cl(F).

The smallest non-trivial closed class of finite graphs is the class C_min of P₄-free graphs, i.e., finite graphs that do not have an induced path on four vertices. This class is generated by the edge and the non-edge. The largest closed class is the class C_max of all finite graphs.

Theorem 4. a) For every clopen graph G on a Polish space, hage(G) is a closed class of finite graphs.

b) For every nontrivial closed class C of finite graphs there is a modular profinite graph G_C of countable weight such that C=hage(G_C)=age(G_C).

c) G_Cmax is a universal modular profinite graph of countable weight.

Theorem 5. a) If G is a clopen graph on a Polish space, then the cochromatic number of G is no larger than the cochromatic number of G_age(G). If G is a self-similar, then the two cochromatic numbers are the same.

b) If G is a clopen graph on a Polish space and age(G) is finitely generated, then the cochromatic number of G and G_Cmin are the same.

These results continue previous research of Goldstern, Kojman, and the author.

Pósa's Conjecture and Related Problems
H. A. Kierstead, Arizona State University
Coauthors: Phong Châu and Louis DeBiasio

Conjecture 1 Every graph G with d(G) ≥ [2/3]|V(G)| contains the square of a hamiltonian cycle.

Pósa's Conjecture remains open, but there has been significant progress. In 1996 G. Fan and the speaker proved a path version of the conjecture. In the same year, Komlós, Sárközy and Szemerédi proved that Pósa's Conjecture, and a more general conjecture of Seymour, hold for all graphs that have sufficiently many vertices. This was one of the first applications of the Regularity-Blow-up Method, and as such, only applies to extremely large graphs. Recently, Levitt, Sárközy and Szemerédi demonstrated a way to avoid the Regularity-Blow-up method. Using some of these ideas as well as older results, Châu, DeBiasio, and the speaker have shown that Pósa's Conjecture holds for all graphs with at least 200 million vertices.

This talk will survey progress on Pósa's Conjecture and related questions.

On the chromatic generalization of the Erdös-Szekeres problem with various numbers of internal points
Vitaliy Koshelev, Moscow Institute of Physics and Technology

The Hajnal-Szemer\' edi Theorem: Extensions and variations
Alexandr Kostochka, University of Illinois, Urbana, USA and Sobolev Institute of Mathematics, Novosibirsk, Russia
Coauthors: Hal Kierstead

An equitable coloring of a graph G is a proper coloring of vertices of G such that the sizes of color classes differ by at most 1. In 1970, Hajnal and Szemer\' edi proved that every graph with maximum degree at most r has an equitable coloring with r+1 colors. This theorem confirmed a conjecture by Erdös and extended a previous result by Corradi and Hajnal. The theorem has numerous applications in combinatorics and other areas of mathematics.

In recent years, there were new results relating to the Hajnal-Szemer\' edi Theorem in the following directions: the Ore-type version of the theorem, the Chen-Lih-Wu Conjecture on equitable colorings of a graph G with D(G) colors, algorithmic versions of the theorem, the list version of the theorem. The aim of the talk is to review the new results on these topics obtained using the approach developed by H. Kierstead and the speaker. This is joint work with H. Kierstead.

The chromatic number of a normed space
Andrey B. Kupavskii, Moscow State University, Mechanics and Mathematics Faculty, Department of Number Theory

Partitions and trees
Jean A Larson, University of Florida

This talk will sketch how to combine the good sequences of Galvin, the tree representations by Schipperus of countable ordinals, and the method of walks by Todorcevic into a representation of countable ordinals useful for proving partition relations, and survey some open problems on partitions.

Delta-Open Sets and Mappings in Topological Spaces
Raja Mohammad Latif, Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Coauthors: Saeid Jafari, College of Vestsjaelland South Herrestraede 11, 4200 Slagelse, Denmark

In 1968 Velicko introduced the concepts of δ-closure and δ-interior operations. We introduce and study properties of δ-derived, δ-border, δ-frontier and δ-exterior of a set using the concept of δ-open sets and study also other properties of the well-known notions of δ-closure and δ-interior. We also introduce some new classes of topological spaces in terms of the concept of δ-D-sets and investigate some of their fundamental properties. Moreover, we investigate and study some further properties of the well-known notions of δ-closure and δ-interior of a set in a topological space. We also introduce δ-R₀ space and study its characteristics. We also introduce δ-continuous, δ-open, δ-irresolute, δ-closed, pre-δ-continuous, pre-δ-open, pre-δ-closed and pre-δ-irresolute mappings and investigate properties and characterizations of these new types of mappings.

References

1.M. Caldas, D.N. Georgian, S. Jafari and T. Noiri, More on δ-semiopen sets, Note di Mathematica, 22(2) (2003/2004), 113-126.

2.J. Cao, M. Ganster, I. Reilly and M. Steiner, δ-closure, θ-closure, and generalized closed sets, Applied General Topology, 6(1) (2005), 79-86.

3.E. Ekici, (δ-pre, s)-continuous functions, Bulletin of the Malaysian Mathematical Sciences Society, Second Series 27(2004), no. 2, 237-251.

4.E. Ekici, On δ-semiopen sets and a generalization of functions, Bol. Soc. Mat. (38) vol. 23 (1-2) (2005), 73-84.

5.A. Kilicman, Z. Salleh, Some results on (δ-pre, s)-continuous functions, International Journal Math. Mat. Sci. 2006(2006), 1-11.

6.T. Noiri, On δ-continuous functions, J. Korean Math. Soc., 16 (1980), 161-166.

7.J.H. Park, B. Y. Lee and M.J. Son, On δ-semiopen sets in topological space, J. Indian Acad. Math., 19(1), (1997), 59-67.

8.S. Raychaudhuri and M.N. Mukherjee, On δ-almost continuity and δ-preopen sets, Bulletin of the Institute of Mathematics, Academia Sinica 21 (1993), no. 4, 357-366.

9.M. Saleh, Some applications of δ-sets to H-closed spaces, Q & A Topology 17(1999), 203-211.

10.M. Saleh, On super and δ-continuities, Mathematics and Mathematics Education, World Scientific, 2002, 281-291.

11.N.V. Velicko, H-closed topological spaces, Mat. Sb., 98-112; English transl. (2), in Amer. Math. Soc. Transl., 78(1968), 102-118.

Abstract

We prove the following

Set and Collection Lemma (2011) If Y is a collection of maximum independent sets of G, then there is a matching

from S --  ć č Ç {A:  A belongs to Y} ö ř into ć č Č {A : A belongs to Y} ö ř  -- S

for every independent set S.

Based on this finding we give alternative proofs for a number of well-known lemmata including :

Clique Collection Lemma (András Hajnal, 1965) If G is a collection of maximum cliques in G, then

ę ę Ç G ę ę ≥ 2•w(G) --   ę ę Č G ę ę .

Maximum Stable Set Lemma (Claude Berge, 1985) An independent set X is maximum if and only if every independent set S disjoint from X can be matched into X.

A generalization of Roitman's theorem for cardinal sequences
Juan Carlos Martínez, Facultat de Matemŕtiques, Universitat de Barcelona
Coauthors: Lajos Soukup

By an LCS space, we mean a locally compact, Hausdorff, scattered space. If X is an LCS space and a is an ordinal, we denote by I_a(X) the a^th-Cantor-Bendixson level of X, i.e. I_a(X) = set of isolated points of X\∪{I_b(X): b < a}. The reduced height of X is defined as ht^-(X) = the least ordinal a such that I_a(X) is finite. Then, the cardinal sequence of X is defined as the sequence formed by the cardinals |I_a(X)| for a < ht^-(X).

Many authors have studied the possible cardinal sequences that can arise as the cardinal sequence of an LCS (equivalently, a superatomic Boolean algebra). By a classical result shown by Roitman in [3], it is consistent with ZFC that there is an LCS X such that ht^-(X) = w₁ + 1, |I_a(X)| = w for every a < w₁ and |I_w1(X)| = w₂.

Then, we shall present the following generalization of Roitman's theorem proved in [2].

Theorem. Suppose that k, l are infinite cardinals such that k⁺⁺ < l, k ^{< k}=k and 2^k = k⁺. Then for each ordinal h with k⁺ ≤ h < k⁺⁺ and cf(h) = k⁺, in some cardinal-preserving generic extension there is an LCS space X such that ht^-(X) = h+ 1, |I_a(X)| = k for every a < h and |I_h(X)| = l.

Several combinatorial notions are needed to prove the above theorem, in particular the notion of a strongly unbounded function introduced by Koszmider in [1].

References

[1] P. Koszmider, Universal matrices and strongly unbounded functions, Mathematical Research Letters 9 (2002), 549-566.

[2] J.C Martínez and L. Soukup, Superatomic Boolean algebras constructed from strongly unbounded functions, preprint.

[3] J. Roitman, A very thin thick superatomic Boolean algebra, Algebra Universalis 21 (1985), 137-142.

Step Up, Here Come Erdös and Hajnal!
János Pach, Rényi Institute and EPFL

It is hard to find any important problem in Ramsey theory to which Erdös and Hajnal did not make a substantial contribution. In this talk, we show how their classic "stepping-up" approach can be applied to the following question. For any sequence of positive integers j₁ < j₂ < ... < j_n, the k-tuples (j_i, j_{i +1}, ..., j_{i + k-1}), i=1, 2, ..., n - k+1, are said to form a monotone path of length n. Given any integers n ≥ k ≥ 2 and q ≥ 2, what is the smallest number N with the property that no matter how we color all k-element subsets of [N]={1, 2, ..., N} with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N_k(q, n), it follows from the 1935 "Happy Ending" paper of Erdös and Szekeres that N₂(q, n)=(n-1)^q+1 and N₃(2, n) = ((2n -4) || (n-2)) + 1. We apply the ideas of Erdös and Hajnal to show that

2(n/q)q-1 ≤ N3(q, n) ≤ 2nq-1logn,

for q ≥ 2 and n ≥ q+2, and to establish analogous bounds on N_k(q, n) for larger values of k. As a geometric application, we prove the following extension of the Happy Ending Theorem. Every family of at least M(n)=2^{n2 logn} plane convex bodies in general position, any pair of which share at most two boundary points, has n members in convex position, that is, it has n members such that each of them contributes a point to the boundary of the convex hull of their union. Joint work with J. Fox, B. Sudakov, and A. Suk.

Forcing related convergences of sequences on complete Boolean algebras
Aleksandar Pavlović, Department of Mathematics and Informatics, University of Novi Sad, Serbia
Coauthors: Miloš S. Kurilić (Department of Mathematics and Informatics, University of Novi Sad, Serbia)

Let convergences l_i : B^w →P(B), 0 ≤ i ≤ 4, on a complete Boolean algebra B be defined in the following way. For a sequence x=〈x_n : n ∈ w〉 in B and the corresponding B-name for a subset of w, let l_i(x)={ ∥ t_x is infinite∥} if b_i(x)=1_B, and empty set otherwise, where b₀ (x)=∥ t_x is finite or cofinite∥, b₁ (x)=∥ t_x is old∥, b₂ (x)=∥ t_x is not unsupported∥, b₃ (x)=∥ t_x is not a splitting real∥ and b₄ (x)=1_B.

Then l₀ is a generalization of the convergence on the Cantor cube, also known as algebraic convergence and it generates the sequential topology on B. On the other hand, l₄ is a generalization of the convergence on the Aleksandrov cube.

Although all convergences are different on each Boolean algebra producing splitting reals, each convergence l_i, 1 ≤ i ≤ 4 generate the same topology. If B is a Maharam algebra, then this topology together with its dual produces the sequential topology, generated by the convergence l₀.

Hypertournaments-scores and losing scores
Shariefuddin Pirzada, King Fahd University of Petroleum and Minerals

We discuss various necesssary and sufficient conditions for the lists of non-negative integers in non-decreasing order to be the losing score lists and score lists of some hypertournament. We also discuss such analogous results for bipartite and multipartite hypertournaments.

Thin coideals
Igor Protasov, Kyiv University

A family H of infinite subsets of w = {0, 1, ...} is coideal if

• M ⊆ N, M ∈ H⇒ N ∈ H;

• M=N₀∪N₁, M ∈ H⇒ N₀ ∈ H∨N₁ ∈ H

In spirit of [1], H is

• P-coideal if, for every decreasing sequence (A_n) in H, there is B ∈ H such that B\A_n is finite for each n.

• Q-coideal if, for every A ∈ H and every partition A=∪_{k ∈ w}F_k with F_k finite, there is B ∈ H, B ⊆ A such that |B∩F_k| < = 1 for each k.

Given a group G of permutations of w, a subset M ⊂ w is called G-thin if, for every g ∈ G, the set

{x ∈ M: gx ≠ x, gx ∈ M}

is finite. We say that a coideal H is thin if, for every countable group G of permutations of w and every M ∈ H, there exists a G-thin subset N ∈ H such that N ⊆ M. A thin ultrafilter is called a T-point in w^*.

Every P-coideal and every Q-coideal is thin. In particular, each P-point and each Q-point in w^* is a T-point. Under CH, there is a T-point which is neither P-, nor Q-point. But I do not know if a T-point can be constructed without additional to ZFC assumptions.

A motivation to study thin coideals is coming from asymptology [2] where thin subsets play the part of convergent sequences.

References

[1] S.Todorcevic. Introduction to Ramsey Spaces, Princeton University Press, 2010

[2] I.Protasov, M.Zarichnyi. General Asymptology, Math. Stud. Monogr. Ser., vol. 12, VNTL Publisher, Lviv, 2007.

On dividing sets into parts of smaller diameters
Andrei M. Raigorodskii, Moscow State University, Mechanics and Mathematics Faculty, Department of Mathematical Statistics and Random Processes; Moscow Institute of Physics and Technology, Faculty of Innovations and High Technology, Department of Data Analysis; Yandex research la
Coauthors: Vera V. Bulankina, Vladislav P. Filimonov, Andrey B. Kupavskiy

On Schur's Theorem on Arithmetic Progressions
Anastasia Rozovskaya, Lomonosov Moscow State University, Faculty of Mechanics and Mathematics Coauthors: Dmitry A. Shabanov, Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Department of Probability Theory, e-mail: shabanov@mech.math.msu.su

On Certain Finite Graphs Without Large Complete and Empty Subgraphs which are Universal
Gábor Sági, Rényi Institute, Budapest

In [] Erdös and Hajnal asked the following question: is it true, that for any finite graph G there exists a constant c=c(G) such that if H is any finite graph on n vertices that does not contain complete or empty subgraphs of size n^c then G is isomorphic to an (induced) subgraph of H? For certain classes of graphs, the answer is known to be true, for recent related results, see e.g. []. However, the original question in its full generality is still open.
We will describe further classes of finite graphs for which the question can be answered affirmatively. Our method is based on the study of natural measures defined on ultraproducts of finite graphs. Particularly, we will show, that (assuming some further technical conditions) if G is a finite graph and H is an ultraproduct of certain special finite graphs equipped with the natural ultraproduct measure such that each complete or empty subgraph of H has measure 0, then G can be embedded into H (as an induced subgraph).

Non-P-points and forcing
Shelah Saharon, The Hebrew University, Rutgers University

We define a property of a (non-principal) ultrafilter on N , i.e. a point of b( N ) very far from P-points. We first under reasonable conditions, prove its existence and its being an ultrafilter being preserved, i.e. continue to generate an ultrafilter, under some forcing and suitable CS iterations. We then prove that consistently such a point generated by ℵ₁ sets exist while no P-point exists, answering a question of Nyikos.

Universal ultrahomogeneous metric spaces
Norbert SAUER, University of Calgary

Universal ultrahomogeneous metric spaces

A metric space M is ultrahomogeneous if for every isometry f of a finite subspace of M to a finite subspace of M there exists an isometry f* of M onto M, (that is, an element of the isometry group of M), which extends f. Let spec(M), the spectrum of M, denote the set of distances between points in M. The metric space M is universal if it is ultrahomogeneous and isometrically embeds every finite metric space F with spec(F) a subset of spec(M).

Results and problems relating to the following questions will be discussed:

1. Given a subset R of the positive reals, does there exists a countable universal metric space M with spec(M)=R.

2. Given a countable universal metric space M and e > 0, does there exists a universal subspace N of M with finite spectrum which e-approximates M?

3. Which universal metric spaces are indivisible or approximately indivisible? (A metric space M is indivisible if for every partition of M into two subspaces there exists an isometric copy of M in one of the two subspaces. A metric space M is approximately indivisible if for every e > 0 and every partition of M into two subspaces there exists an isometric copy of M at distance at most e from one of the two subspaces.)

4. Are completions of universal metric spaces universal?

5. The relationship between divisibility notions for metric spaces and notions of topological dynamics such as oscillation stability and extremely amenable groups.

J. Lopez-Abad and L. Nguyen Van Thé, The oscillation stability problem for the Urysohn sphere: A combinatorial approach, Topology Appl. 155 Issue 14 (2008) 1516-1530.

L. Nguyen Van Thé, N. Sauer, The Urysohn Sphere is oscillation stable, Goemetric and Functional Analysis 19 Issue 2 (2009) 536-557.

N. Sauer, Vertex partitions of metric spaces with finite distance sets, Discrete Mathematics submitted.

N. Sauer, Approximating universal metric spaces, manuscript.

The diameter of symmetric groups
Ákos Seress, The Ohio State University and The University of Western Australia

For a finite group G, the diameter of G is defined as the maximum of the diameters of all Cayley graphs of G. A 25 year-old conjecture states that the diameter of the full symmetric group S_n is polynomial in n.

We shall review the attempts to prove this conjecture. Although the formulation of the problem is algebraic, the methods are mostly combinatorial.

On the Erdös - Hajnal problem in the case of three colors
Dmitry A. Shabanov, Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Department of Probability Theory

A game on Boolean algebras
Boris Sobot, Department of Mathematics and Informatics, University of Novi Sad, Serbia
Coauthors: Milo? Kurilić, Department of Mathematics and Informatics, University of Novi Sad, Serbia

The game G_ls (k) is played on a complete Boolean algebra B, by two players, White and Black, in k-many moves (where k is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ B. In the a-th move White chooses p_a ∈ (0, p)_B and Black responds choosing i_a ∈ {0, 1}. White wins the play iff ∧_{b ∈ k}∨_{a ≥ b}p_a^i_a=0, where p_a⁰=p_a and p_a¹=p\p_a.

The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if k ≥ p(B) or if B contains a k⁺-closed dense subset. On the other hand, if White has a w.s., then k ∈ [h₂(B), p(B)). The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2 ^{< k}=k ∈ Reg and forcing by B preserves the regularity of k, then White has a w.s. iff the power 2^k is collapsed to k in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. B such that White (respectively, Black) has a w.s. for each infinite cardinal k ∈ S (resp. k ∉ S). Also it is shown consistent that for each k ∈ Reg there is a c.B.a. on which the game G_ls (k) is undetermined.

Rainbow Colourings
Lajos Soukup, Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences

Given a function f:[X]ⁿ→ X we say that a subset Y ⊂ X is a rainbow subset for f provided that f\restriction[Y]ⁿ is injective. A typical question was a problem of Erdös: Assume that a colouring f:[w₁]²→ 3 establishes the partition relation w₁\not→[w₁]²₃. Is there a rainbow triangle for f?

We survey some old results and presents some theorems.

Long Arithmetic Progressions in Sumsets
Endre Szemerédi, Alféd Rényi Institute of Mathematics

Let A ⊂ [1, n] be a set of integers. Let S(A) = {a_i1+a_i2+...+a_ik; a_ij ∈ A } be the set of sums of subsets of A. It has been proved that there exists a large c such that if |A| > c √n then S(A) contains an arithmetic progression of length n. We are going to prove that if |A| > 4 √n then S(A) contains an arithmetic progression of length n. We are going to try to describe some ideas how 4 might be reduced to 2 which would be the optimal value.

On densest sets avoiding unit distance in spaces of small dimension
Mariya V. Titova, Moscow State University, Mechanics and Mathematics Faculty, Department of Mathematical Statistics and Random Processes
Coauthors: Andrei B. Kupavskii, Andrei M. Raigorodskii

A combinatorial analysis of the uncoditional basic sequence problem
Stevo Todorcevic
Coauthors: J. Lopez-Abad

We analyze properties of cardinals k that guarantee that normed spaces of density k must contain infinite basic unconditional sequence.

Stationary reflection and two cardinal tree properties
Boban Velickovic
Department of Mathematics, University of Paris 7.

Reflection principles are a way of transfering large cardinal properties to small cardinals. We consider several versions of stationary and semi stationary reflection and their consequences. In particular, we show that semi stationary set reflection implies the Singular Cardinal Hypothesis, the failure of weak square, etc. We also investigate a connection with two cardinal tree properties which were recently introduced by Weiss and prove that it follows from stationary or semi stationary reflection augmented with a weak form of Martin's Axiom. Using the approach recently developed by Viale and Weiss we conclude that any standard method to obtain (semi) stationary set reflection together with MA(Cohen) requires at least a strongly compact cardinal.

This is joint work with H. Sakai (Kobe University).

Asymmetric partition relations in the countably infinite case.
Thilo Weinert, Hausdorff Research Institute for Mathematics

A characterization of a partition relation between finite multiples of omega^2, finite multiples of omega^2 and natural numbers is given. Additionally we are going to discuss the simplest example, R(omega^2 * 2, 3), i.e. the minimal ordinal number alpha such that alpha --> (omega^2 * 2, 3)^2 holds true.

Borel Conjecture and dual Borel Conjecture (and other variants of the Borel Conjecture)
Wolfgang Wohofsky, Vienna University of Technology
Coauthors: Martin Goldstern (Vienna University of Technology), Jakob Kellner (KGRC, University of Vienna), Saharon Shelah (Hebrew University of Jerusalem; Rutgers University)

The Borel Conjecture (BC) says that there are no uncountable strong measure zero sets, whereas the dual Borel Conjecture (dBC) is the analogous statement about strongly meager sets. In 1976, Laver showed the consistency of BC; Carlson showed that dBC is consistent.

Together with Martin Goldstern, Jakob Kellner and Saharon Shelah, I have been working on the following theorem: It is consistent that BC and dBC hold simultaneously. In my talk, I will give an overview of the proof.

Moreover, I will discuss another variant of the Borel Conjecture, which I call the Marczewski Borel Conjecture (MBC). It says that there are no uncountable sets which can be translated away from each Marczewski null set. Is MBC consistent? I will present a connection between MBC and the class of translation-invariant sigma-ideals which are dense in Sacks forcing (which I call "Sacks dense ideals").

Website: http://www.wohofsky.eu/math/

On connectivity of random distance graphs of special type
Andrey R. Yarmukhametov, Moscow State University, Mechanics and Mathematics Faculty, Department of Mathematical Statistics

On zero-one laws for random graphs
Mаksim E. Zhukovskii, Moscow State University, Mechanics and Mathematics Faculty, Department of Probability Theory; Moscow Institute of Physics and Technology, Faculty of Innovations and High Technology, Department of Data Analysis