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Iteration of Semiproper Forcing Revisited [article]

Boban Velickovic
2014 arXiv   pre-print
We present a method for iterating semiproper forcing which uses side conditions and is inspired by the technique recently introduced by Neeman.
arXiv:1410.5095v1 fatcat:xltsoja7s5ewzi73d4kvkjvoju

Non-vanishing higher derived limits [article]

Boban Velickovic, Alessandro Vignati
2021 arXiv   pre-print
In the study of strong homology Mardešić and Prasolov isolated a certain inverse system of abelian groups 𝐀 indexed by elements of ω^ω. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits lim^n 𝐀 must vanish, for n>0. They also proved that under the Continuum Hypothesis lim^1 𝐀≠ 0. The question whether lim^n 𝐀 vanishes, for n>0, has attracted considerable interest from set theorists. Dow, Simon and
more » ... han showed that under PFA lim^1 𝐀 =0. Bergfalk show that it is consistent that lim^2𝐀 does not vanish. Later Bergfalk and Lambie-Hanson showed that, modulo a weakly compact cardinal, it is relatively consistent with ZFC that lim^n 𝐀 =0, for all n. The large cardinal assumption was recently removed by Bergfalk, Hrušak and Lambie-Henson. We complete the picture by showing that, for any n>0, it is relatively consistent with ZFC that lim^n 𝐀≠ 0.
arXiv:2107.03787v1 fatcat:zut5agckuvb55pcnk3ngy63ciy

PFA and precipitousness of the nonstationary ideal [article]

Boban Velickovic
2015 arXiv   pre-print
We apply Neeman's method of forcing with side conditions to show that PFA does not imply the precipitousness of the nonstationary ideal on ω_1.
arXiv:1504.05326v1 fatcat:kr4cqftlhrcrxmgoutchpqe5pe

Oscillations and their applications in partition calculus [article]

Laura Fontanella, Boban Velickovic
2011 arXiv   pre-print
The following theorem is due to Veličković and Woodin ( [11] ). Theorem 3 ([11]). Let X, Y, Z ⊆ [N] ω be superperfect sets. Then o ′′ [X × Y × Z] ⊇ 2 ω . Proof.  ... 
arXiv:1110.0614v1 fatcat:52ld5hrypjeybkwo52ui5lfype

Ranks of Maharam algebras [article]

Zikica Perovic, Boban Velickovic
2016 arXiv   pre-print
Solving a well-known problem of Maharam, Talagrand [17] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This
more » ... s rise to uncountably many non isomorphic separable atomless Maharam algebras.
arXiv:1608.02468v2 fatcat:aqjoyigoa5en3fh53zvcl5jo6i

Positional strategies in long Ehrenfeucht-Fraissé games [article]

Saharon Shelah, Jouko Väänänen, Boban Velickovic
2013 arXiv   pre-print
We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fra\"iss\'e-game of length \omega_1 but there is no \sigma-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.
arXiv:1308.0156v1 fatcat:iqxsarb4lfhrzfrxr7khkglgxm

Proper forcing remastered [article]

Boban Velickovic, Giorgio Venturi
2013 arXiv   pre-print
In these notes we present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: the Friedman-Mitchell poset for adding a club in \omega_2 with finite conditions, Koszmider's forcing construction of a strong chain of length \omega_2 of functions from \omega_1 to \omega_1, and the Baumgartner-Shelah forcing construction of a thin very tall superatomic Boolean algebra.
arXiv:1110.0610v3 fatcat:agafxndjjbhxvnlw7h64urpg2i

Forcing axioms and stationary sets

Boban Velic̆ković
1992 Advances in Mathematics  
BOBAN VELICKOVIC LEMMA 2.4. Vy + SPFA. Proof Let 9 in VY be a poset which preserves stationary subsets of w1 and let (D, : a < o, ) be names for dense subsets of 9.  ...  BOBAN VELICKOVk CLAIM. B * 9 preserves stationary subsets of ml. Let n: 9 *3 + 9 be the natural projection. Set 9*= {n-'(D):DELB}.  ... 
doi:10.1016/0001-8708(92)90038-m fatcat:old5a3ct5fasphgwuuhfzr24ya

A note on Tsirelson type ideals

Boban Veličković
1999 Fundamenta Mathematicae  
Using Tsirelson's well-known example of a Banach space which does not contain a copy of c 0 or lp, for p ≥ 1, we construct a simple Borel ideal I T such that the Borel cardinalities of the quotient spaces P(N)/I T and P(N)/I 0 are incomparable, where I 0 is the summable ideal of all sets A ⊆ N such that n∈A 1/(n + 1) < ∞. This disproves a "trichotomy" conjecture for Borel ideals proposed by Kechris and Mazur.
doi:10.4064/fm-159-3-259-268 fatcat:p4ehbejnonc2jfsrhy4gtsbnzu

Maharam algebras

Boban Veličković
2009 Annals of Pure and Applied Logic  
Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam in [24] in relation to Von Neumann's problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the main open problem in this area for around 60 years. It was finally resolved by Talagrand [31] who provided the first example of a Maharam algebra which is not a measure algebra. In this paper we
more » ... ey some recent work on Maharam algebras in relation to the two conditions proposed by Von Neumann: weak distributivity and the countable chain condition. It turns out that by strengthening either one of these conditions one obtains a ZFC characterization of Maharam algebras. We also present some results on Maharam algebras as forcing notions showing that they share some of the well known properties of measure algebras. 1. B has the countable chain condition (ccc), i.e. if A ⊆ B is such that a∧b = 0, for every a, b ∈ A such that a = b, then A is at most countable. 2. B is weakly distributive, i.e. if {b n,k } n,k is a double sequence of elements of B then the following weak distributivity law holds:
doi:10.1016/j.apal.2008.04.006 fatcat:6zpp4hcw3vgsppg4k2pnazhhhm

Playful Boolean Algebras

Boban Velickovic
1986 Transactions of the American Mathematical Society  
We show that for an atomless complete Boolean algebra 8 of density < 2N°, the Banach-Mazur, the split and choose, and the Ulam game on S are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff B adds a real; Nonempty wins iff B has a (r-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáá.
doi:10.2307/2000386 fatcat:r6lgmhiqvjb2ppou3j3jwzsmke

Guessing models and the approachability ideal [article]

Rahman Mohammadpour, Boban Velickovic
2019 arXiv   pre-print
Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call GM^+(ω_3,ω_1) holds. This principle implies ISP(ω_2) and ISP(ω_3), and hence the tree property at ω_2 and ω_3, the Singular Cardinal Hypothesis, and the failure of the weak square principle (ω_2,λ), for all regular λ≥ω_2. In addition, it implies that the restriction of the approachability ideal I[ω_2] to the set of ordinals of cofinality ω_1 is the non stationary ideal on
more » ... s set. The consistency of this last statement was previously shown by Mitchell.
arXiv:1802.10125v2 fatcat:c3nduoznjzcypc4fud5pvdzluu

CCC forcing and splitting reals

Boban Velickovic
2005 Israel Journal of Mathematics  
Prikry asked if it is relatively consistent with the usual axioms of ZFC that every nontrivial ccc forcing adds either a Cohen or a random real. Both Cohen and random reals have the property that they neither contain nor are disjoint from an infinite set of integers in the ground model, i.e. they are splitting reals. In this note I show that that it is relatively consistent with ZFC that every non atomic weakly distributive ccc forcing adds a splitting real. This holds, for instance, under the
more » ... roper Forcing Axiom and is proved using the P -ideal dichotomy first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. In the process,
doi:10.1007/bf02785365 fatcat:hqvowqwmgrgstfdf6wfb6t6rua

The complexity of the reals in inner models of set theory [article]

Boban Velickovic, W. Hugh Woodin
1995 arXiv   pre-print
The usual definition of the set of constructible reals is Σ ^1_2. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be analytic or even Borel in a nontrivial way. A related problem was posed by K. Prikry: can there exist a nonconstructible perfect set of constructible reals? The main result of this paper is a negative answer to Friedman's question. In fact we prove that if M is an
more » ... er model of set theory and the set of reals in M is analytic then either all reals are in M or else _1^M is countable. We also extend this result to higher levels of the projective hierarchy under appropriate large cardinal assumptions. Concerning Prikry's problem we show that the answer is negative if "perfect" is replaced by "superperfect" but that it can be positive if "constructible" is replaced by "belonging to some inner model M".
arXiv:math/9501203v1 fatcat:e44xbwvpzvcgzlmfmytkcjckvm

Borel partitions of infinite subtrees of a perfect tree [article]

Alain Louveau, Boban Veličković, Saharon Shelah
1993 arXiv   pre-print
A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n >= 3 is false. Let us identify the reals with 2^omega ordered by the lexicographical ordering and define for distinct x,y in 2^omega, D(x,y) to be the least n such that x(n) not= y(n). Let the type of an increasing n-tuple x_0, ... x_n-1_< be the ordering <^* on 0,
more » ... ,n-2 defined by i<^*j iff D(x_i,x_i+1)< D(x_j,x_j+1). Galvin proved that for any Borel coloring of triples of reals there is a perfect set P such that the color of any triple from P depends only on its type. Blass proved an analogous result is true for any n. As a corollary it follows that if the unordered n-tuples of reals are colored into finitely many Borel classes there is a perfect set P such that the n-tuples from P meet at most (n-1)! classes. We consider extensions of this result to partitions of infinite increasing sequences of reals. We show, that for any Borel or even analytic partition of all increasing sequences of reals there is a perfect set P such that all strongly increasing sequences from P lie in the same class.
arXiv:math/9301209v1 fatcat:hj76wbl7brfcdi7jz6qfq5jfeq
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