Subject / 9 records

Combinatorics and Ramsey theory

Posets, finite configurations, Ramsey phenomena, vector balancing, and combinatorial obstructions.

Classification: Second Brain

9 records

• 10.5281/zenodo.238333062026

  Finite configurations and nuclear quotients of Thompson's group F

  Martino Lupini · Thompson groups, C*-algebras, Operator algebras

  Let F be Thompson’s group, let P = F′, and let Ug denote the canonical prefix-table unitaries in the Cuntz algebra O₂. We prove that the finite ordinary tensor images Bn = C*(Uf⊗n : f ∈ F) and Cn = C*(Up⊗n : p ∈ P) are nuclear. A representation has finite separated derived capacity when products of sufficiently many displacement operators with separated interior supports vanish. Capacity at most d gives an actual marked quotient of B2d+2, or of C2d for a nontrivial P-representation. Infinite capacity is equivalent to admitting the regular representation as an actual quotient. Consequently, if F is nonamenable, every nonzero nuclear quotient of its full group C*-algebra, and likewise of C*(P), is a quotient of one of these finite canonical images. The finite nuclearity and norm estimates are unconditional; nonamenability enters only in passing from nuclearity to finite capacity. The result gives a countable cofinal family of nuclear quotients, with a degree depending on the quotient.

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• 10.5281/zenodo.234611402026

  Endomorphism counts from primitive orbits and synchronized block actions

  Martino Lupini · Orderability, Posets, Groups

  For a finite poset P, let R(P) = |End(P)|/|Aut(P)|, where endomorphisms preserve the reflexive order. We prove R(P) ≥ 2^{|P|/(64 log₂ |P|)} for |P| ≥ 4 when every chain has at most two elements and the action on each nonsingleton automorphism orbit is primitive. The same bound holds for a class of imprimitive actions: on each interaction component the action must be the full automorphism group of a rooted tree, acting identically on all the identified orbits. The proof combines simultaneous collapses of indices with maps into principal ideals and filters. We give counting lemmas that retain the whole kernel of a block action, and a quantitative collapse, degree, or orbit-size alternative for normal products of alternating groups with natural orbits. We also construct connected twin-free posets with group S₃ wr Sₜ, constant minimum of the two maximum principal-set sizes, and imprimitive action on every orbit, although every orbit-preserving endomorphism is an automorphism. A restricted three-level extension has lower bound 2^{|P|/(128 (log₂ |P|)²)}. These results give partial answers to the finite-poset automorphism problem of Rival and Rutkowski; the unrestricted question remains beyond the arguments here.

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• 10.5281/zenodo.234584502026

  A countable scattered poset without a strongly maximal chain

  Martino Lupini · Orderability, Posets

  A chain C in a partially ordered set is strongly maximal if |D \ C| ≤ |C \ D| for every chain D. We construct a countably infinite poset with no infinite antichain and no strongly maximal chain. The poset is scattered: none of its chains contains a copy of the rational order. This gives a negative answer, already in the countable scattered case, to the strong-chain question posed by Hollom in 2024 and restated in 2026. The construction replaces each point (x, y, n) of Hollom’s layered poset by a finite chain of length 2^x + 2^y. A finite path exchange restricts the possible chains, after which a further finite exchange always improves them. We also analyze a width-two example of Hollom showing that the chain-extension inclusion used in an earlier claimed positive proof need not hold.

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• 10.5281/zenodo.234527882026

  The small index property for the random poset

  Martino Lupini · Posets, Descriptive set theory, Groups

  Let P be the countable universal homogeneous poset. We prove that every abstract subgroup H ≤ Aut(P) with [Aut(P):H] < 2^{ℵ₀} contains the pointwise stabilizer of a finite subset of P. Thus Aut(P) has the small index property, answering the question recorded as open by Barbina and Macpherson in 2005. No regularity assumption on H is required. The proof equips P with a generic strictly increasing rational-valued rank. Its stabilizer has ample generics, and generic pairs of ranks yield a comeager double coset. We show that the small index property passes from a closed subgroup with such a double coset to the ambient Polish group.

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• 10.5281/zenodo.234401542026

  Common null complements for polynomial images of finite sums of perfect sets

  Martino Lupini · Mathematics

  Fix an integer D ≥ 1, and let A₁, …, A₍D₊₂₎ be nonempty compact perfect subsets of the real line. For their Minkowski sum C = A₁ + ⋯ + A₍D₊₂₎, we construct a single null Fσ set B ⊆ ℝ such that P(C) + B = ℝ for every nonconstant real polynomial P of degree at most D. In particular, every sum of three compact perfect sets has one null complement that works under every nonzero dilation. Thus the conclusion of Jung and Lai’s Conjecture 4.8 holds for these sums; the conjecture remains open for arbitrary perfect sets. The conclusion applies to some sums with zero logarithmic Hausdorff and packing dimensions that contain no polynomial image of an infinite summable finite-digit product. The proof isolates a local sum hypothesis that permits the same construction for more general compact sets.

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• 10.5281/zenodo.233720662026

  Finite sums and idempotency for dynamically central piecewise syndetic sets

  Martino Lupini · Mathematics

  A set of positive integers is a set of pointwise recurrence if it supplies a return of every point to every neighborhood of itself in every minimal topological dynamical system. We prove that every such set contains finite-sums configurations of every finite rank. More generally, this holds after intersection with a return set to a neighborhood of a point in a minimal system: every dynamically central piecewise syndetic set is IP₀. The finite-sums configurations can be chosen with a piecewise syndetic set of simultaneous completions; their generators may be taken superincreasing.

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• 10.5281/zenodo.231050332026

  An eight-element parachute lattice and its dual: finite representations with two unary operations

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.229325952026

  Lattice-Ordered Groups as Full Automorphism Groups of Posets

  Martino Lupini · Orderability, Posets, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227583052026

  Circular colorings of abelian Cayley graphs below four

  Martino Lupini · Mathematics

  No abstract on this deposit.

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