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Borel Partitions of Products of Finite Sets and the Ackermann Function

Carlos A. Di Prisco, Jimena Llopis, Stevo Todorcevic
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
It is shown that for every primitive recursive sequence [m i ] i=0 of positive integers, there is an ackermannic sequence [n i ] i=0 of positive integers such that for every partition of the product >  ...  i=0 n i into two Borel pieces, there are sets H i n i with |H i |=m i such that the subproduct > i=0 H i is included in one of the pieces.  ...  A BASIC RESULT FOR CLOSED SETS The following lemma shows that there is a very uniform way to get finite solutions for partition relations of finite products of finite sets. Proof.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jcta.2000.3082">doi:10.1006/jcta.2000.3082</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nduqbikezndrjatknvzvr5qqxa">fatcat:nduqbikezndrjatknvzvr5qqxa</a> </span>
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Page 8593 of Mathematical Reviews Vol. , Issue 2001M [page]

<span title="">2001</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(YV-IVIC; Caracas) ; Llopis, Jimena (YV-SBOL; Caracas); Todorcevic, Stevo (YU-BELG-IM; Belgrade) Borel partitions of products of finite sets and the Ackermann function. (English summary) J. Combin.  ...  Grzegorek introduced two subclasses of AFC: (1) sets A such that every Borel isomorphic image of A in X is meager in ¥ and (2) sets A that do not contain any one-to-one Borel image of a non-meager subset  ... 
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Souslin partitions of products of finite sets

C.A. Di Prisco, S. Todorcevic
<span title="">2003</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/37jjomjmvfhrzf3di2gjnrrfuu" style="color: black;">Advances in Mathematics</a> </i> &nbsp;
theory is based on the fact that for every CLðm mÞ-measurable partition for every PCLðm mÞ-measurable partition c : o o ½o o -2 there is HA½o o and a sequence fH i : iAog of sets with H i Do and jH i  ...  A 93 (2001) 333; Combinatorica, to appear) it is shown that CLðm mÞ and PCLðm mÞ are s-fields that contain all closed sets, and therefore all Borel subsets of their corresponding domains.  ...  a subfamily c x ; ðxAPÞ of a given family c x ; ðxAo o Þ of partitions of ½o o ; the set P will essentially have to be included in a product of finite sets.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0001-8708(02)00064-6">doi:10.1016/s0001-8708(02)00064-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4hbym4rnqbh3fjosdwltsnuwpi">fatcat:4hbym4rnqbh3fjosdwltsnuwpi</a> </span>
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Polarized partitions on the second level of the projective hierarchy

Jörg Brendle, Yurii Khomskii
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
In this paper we investigate these properties on the 1 2 -and 1 2 -levels of the projective hierarchy, i.e., we investigate the strength of the statements "all 1 2 / 1 2 sets satisfy the (bounded) polarized  ...  Di Prisco and Todorčević (2003) [6] proved that both partition properties are true for analytic sets A.  ...  Acknowledgments The authors would like to thank Jindřich Zapletal for drawing their attention to [17] and the forcing contained therein, which significantly influenced the content of Section 5. The  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2012.03.003">doi:10.1016/j.apal.2012.03.003</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j4wwvq6ek5gmjj77k2xzqtmaqe">fatcat:j4wwvq6ek5gmjj77k2xzqtmaqe</a> </span>
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Laver and set theory

Akihiro Kanamori
<span title="2016-01-18">2016</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3egevcnserbbni4tcigsztdici" style="color: black;">Archive for Mathematical Logic</a> </i> &nbsp;
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.  ...  Doctoral students of Richard Laver Stephen Grantham, An analysis of Galvin In addition to having these doctoral students at Boulder, Laver was on the thesis committees of, among many:  ...  A decade later Laver [80] pursued the study the space of functions : ω 1 → ω 1 under eventual dominance modulo finite sets.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00153-015-0462-7">doi:10.1007/s00153-015-0462-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ziwvh27arbprhsdlmku4f2npe">fatcat:5ziwvh27arbprhsdlmku4f2npe</a> </span>
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Entropy and isoperimetry for linear and non-linear group actions

Misha Gromov
<span title="">2008</span> <i title="European Mathematical Publishing House"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/abuyydqk4bhulofdonouszmati" style="color: black;">Groups, Geometry, and Dynamics</a> </i> &nbsp;
(and/or measures) of sets and of the dimensions of linear spaces.  ...  We prove inequalities of isoperimetric type for groups acting on linear spaces and discuss related geometric and combinatorial problems, where we use the Boltzman entropy to keep track of the cardinalities  ...  Finite Example. Let P be a partition of X, a finite set with the unitary atoms and take a subset Y ⊂ X.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4171/ggd/48">doi:10.4171/ggd/48</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6wseg4dbwfgupmc4r57lipbram">fatcat:6wseg4dbwfgupmc4r57lipbram</a> </span>
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Equipartition of several measures [article]

R.N. Karasev
<span title="2013-06-14">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Another example is: Any convex body in the plane can be partitioned into q parts of equal areas and perimeters provided q is a prime power.  ...  A newer version of this paper, merged with the similar paper of A. Hubard and B. Aronov is arXiv:1306.2741.  ...  The author thanks Arseniy Akopyan, Boris Aronov, Pavle Blago-  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1011.4762v7">arXiv:1011.4762v7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yieync6b7fanvbkvfesxc2dru4">fatcat:yieync6b7fanvbkvfesxc2dru4</a> </span>
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Set Theory from Cantor to Cohen [chapter]

Akihiro Kanamori
<span title="">2012</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3htcvojut5de5ct6zf42o424vi" style="color: black;">Handbook of the History of Logic</a> </i> &nbsp;
into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory.  ...  Set theory is a particular case of a field of mathematics in which seminal proofs and pivotal problems actually shaped the basic concepts and forged axiomatizations, these transmuting the very notion of  ...  and appears here with the permission of the Bulletin.  ... 
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SET THEORY FROM CANTOR TO COHEN [chapter]

Akihiro Kanamori
<span title="">2009</span> <i title="Elsevier"> Philosophy of Mathematics </i> &nbsp;
into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory.  ...  Set theory is a particular case of a field of mathematics in which seminal proofs and pivotal problems actually shaped the basic concepts and forged axiomatizations, these transmuting the very notion of  ...  and appears here with the permission of the Bulletin.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/b978-0-444-51555-1.50014-6">doi:10.1016/b978-0-444-51555-1.50014-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vnj2vdlpofdp7pg37tyiemdkiq">fatcat:vnj2vdlpofdp7pg37tyiemdkiq</a> </span>
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Well Quasi-Orders in Computer Science (Dagstuhl Seminar 16031)

Jean Goubault-Larrecq, Monika Seisenberger, Victor Selivanov, Andreas Weiermann, Marc Herbstritt
<span title="2016-07-13">2016</span> <i title="Schloss Dagstuhl Leibniz-Zentrum für Informatik GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/u2phfyhrhje7xnnubtp3vfklcu" style="color: black;">Dagstuhl Reports</a> </i> &nbsp;
, topological complexity of computational problems on continuous functions).  ...  This report documents the program and the outcomes of Dagstuhl Seminar 16031 "Well Quasi-Orders in Computer Science", the first seminar devoted to the multiple and deep interactions between the theory  ...  Wagner hierarchy from sets of finite partitions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/dagrep.6.1.69">doi:10.4230/dagrep.6.1.69</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/journals/dagstuhl-reports/Goubault-Larrecq16.html">dblp:journals/dagstuhl-reports/Goubault-Larrecq16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o3uwzu5ptfavfes6kgexpi4a6q">fatcat:o3uwzu5ptfavfes6kgexpi4a6q</a> </span>
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The Rado Simplicial Complex [article]

Michael Farber, Lewis Mead, Lewin Strauss
<span title="2020-01-30">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also prove several other interesting properties of the Rado complex X, for example we show that removing any finite set of simplexes of X gives a complex isomorphic to X.  ...  The geometric realisation |X| of a Rado complex is contractible and is homeomorphic to an infinite dimensional simplex.  ...  The paper [2] studies the Fraïssé limit of the class of all finite simplicial complexes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1912.02515v2">arXiv:1912.02515v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uxgltsr7bjgbnonj6jum3puvnm">fatcat:uxgltsr7bjgbnonj6jum3puvnm</a> </span>
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On the ℓ-modular composition factors of the Steinberg representation [article]

Meinolf Geck
<span title="2015-04-20">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We give a new proof of this result in terms of the Hecke algebra of G with respect to a Borel subgroup and show how to identify the simple socle of _k among the principal series representations of G.  ...  Let G be a finite group of Lie type and _k be the Steinberg representation of G, defined over a field k. We are interested in the case where k has prime characteristic ℓ and _k is reducible.  ...  I wish to thank Gerhard Hiss for clarifying discussions about the contents of [6] , [16] , [17] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1504.04157v2">arXiv:1504.04157v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jrovls6t5vep7ih2m2cvr5rszy">fatcat:jrovls6t5vep7ih2m2cvr5rszy</a> </span>
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The Development of Mathematical Logic from Russell to Tarski, 1900–1935 [chapter]

Paolo Mancosu, Richard Zach, Calixto Badesa
<span title="2009-06-18">2009</span> <i title="Oxford University Press"> The Development of Modern Logic </i> &nbsp;
Consider the set M. By the power set axiom we can form U(M), then U(U(M)) and so on for any finite iteration of the power set axiom.  ...  Von Neumann takes the notion of function as basic (the notion of set can be recovered from that of function) and allows classes in addition to sets.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/acprof:oso/9780195137316.003.0029">doi:10.1093/acprof:oso/9780195137316.003.0029</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ygiplet5oraf7fza6fsmf5j5ly">fatcat:ygiplet5oraf7fza6fsmf5j5ly</a> </span>
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Convex equipartitions: the spicy chicken theorem

Roman Karasev, Alfredo Hubard, Boris Aronov
<span title="2013-07-24">2013</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5kcmcgmosjd2tbsrk5wnkph4j4" style="color: black;">Geometriae Dedicata</a> </i> &nbsp;
We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into n convex sets with equal volumes and equal surface areas.  ...  its icing, and a generalization of the Gromov-Borsuk-Ulam theorem for convex sets in the model spaces of constant curvature.  ...  We also thank an anonymous referee for encouraging us to merge our papers and for his/her enthusiasm towards the chicken nuggets description of Corollary 1.1.  ... 
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The Rado simplicial complex

Michael Farber, Lewis Mead, Lewin Strauss
<span title="2021-05-06">2021</span> <i title="Springer Science and Business Media LLC"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c5jvinnaofg2pnrpaxbxvqkocq" style="color: black;">Journal of Applied and Computational Topology</a> </i> &nbsp;
We prove that X is the unique simplicial complex which is both universal and homogeneous. The 1-skeleton of X is the well-known Rado graph.  ...  We observe several curious properties of X, for example we show that X is robust, i.e. removing any finite set of simplexes leaves a simplicial complex isomorphic to X.  ...  If the vertex set V (X ) is partitioned into a finite number of parts then the induced subcomplex on at least one of these parts is a Rado complex.  ... 
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