Set-theoretic constructions of nonshrinking open covers

On The Modal Logics of Some Set-Theoretic Constructions

Approximating set multi-covers

Lovász and Stein (independently) proved that any hypergraph satisfies τ ≤ (1+ln ∆)τ * , where τ is the transversal number, τ * is its fractional version, and ∆ denotes the maximum degree. We prove τ f ≤ 3.17τ * max{ln ∆, f } for the f-fold transversal number τ f. Similarly to Lovász and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm. As a com...

Combinatorics of open covers (VIII)∗

For each space, Ufin(Γ,Ω) is equivalent to Sfin(Ω,O) and this selection property has game-theoretic and Ramsey-theoretic characterizations (Theorem 2). For Lindelöf space X we characterize when a subspace Y is relatively Hurewicz in X in terms of selection principles (Theorem 9), and for metrizable X in terms of basis properties, and measurelike properties (Theorems 14 and 16). Using the Contin...

Rigidity-Theoretic Constructions of Integral Fary Embeddings

Fáry [3] proved that all planar graphs can be drawn in the plane using only straight line segments. Harborth et al. [7] ask whether or not there exists such a drawing where all edges have integer lengths, and Geelen et al. [4] proved that cubic planar graphs satisfied this conjecture. We re-prove their result using rigidity theory, exhibit other natural families of planar graphs that satisfy th...

Model-theoretic constructions via amalgamation and reducts

Recall that a complete first-order theory with infinite models is strongly minimal if in any of its models, every parameter-definable subset of the model is finite or cofinite. Classical examples are theories of vector spaces and algebraically closed fields; also the degenerate example of the theory of infinite ‘pure’ sets where the only structure comes from equality. Algebraic closure in a str...