@article{18559,
  abstract     = {We prove a 1973 conjecture due to Erdős on the existence of Steiner triple systems with arbitrarily high girth.},
  author       = {Kwan, Matthew Alan and Sah, Ashwin and Sawhney, Mehtaab and Simkin, Michael},
  issn         = {1939-8980},
  journal      = {Annals of Mathematics},
  number       = {3},
  pages        = {1059--1156},
  publisher    = {Princeton University},
  title        = {{High-girth Steiner triple systems}},
  doi          = {10.4007/annals.2024.200.3.4},
  volume       = {200},
  year         = {2024},
}

@article{2357,
  abstract     = {The classic Poincaré inequality bounds the L q-norm of a function f in a bounded domain Ω ⊂ ℝ n in terms of some L p-norm of its gradient in Ω. We generalize this in two ways: In the first generalization we remove a set Τ from Ω and concentrate our attention on Λ = Ω \ Τ. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of Τ is arbitrarily small. A Poincaré inequality does hold, however, if one makes the additional assumption that f has a finite L p gradient norm on the whole of Ω, not just on Λ. The important point is that the Poincaré inequality thus obtained bounds the L q-norm of f in terms of the L p gradient norm on Λ (not Ω) plus an additional term that goes to zero as the volume of Τ goes to zero. This error term depends on Τ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the 'nice' domain Ω. In the second generalization we are given a vector field A and replace ∇ by ∇ + iA(x) (geometrically, a connection on a U(1) bundle). Unlike the A = 0 case, the infimum of ∥(∇ + iA)f∥ p over all f with a given ∥f∥ q is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.},
  author       = {Lieb, Élliott and Seiringer, Robert and Yngvason, Jakob},
  issn         = {1939-8980},
  journal      = {Annals of Mathematics},
  number       = {3},
  pages        = {1067 -- 1080},
  publisher    = {Princeton University Press},
  title        = {{Poincaré inequalities in punctured domains}},
  doi          = {10.4007/annals.2003.158.1067 },
  volume       = {158},
  year         = {2003},
}