@article{21159,
  abstract     = {One of the foundational theorems of extremal graph theory is Dirac’s theorem, which
says that if an n-vertex graph G has minimum degree at least n/2, then G has a
Hamilton cycle, and therefore a perfect matching (if n is even). Later work by Sárközy,
Selkow and Szemerédi showed that in fact Dirac graphs have many Hamilton cycles
and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise
description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph
G (in terms of an entropy-like parameter of G). In this paper we extend Cuckler
and Kahn’s result to perfect matchings in hypergraphs. For positive integers d < k,
and for n divisible by k, let md (k, n) be the minimum d-degree that ensures the
existence of a perfect matching in an n-vertex k-uniform hypergraph. In general, it is
an open question to determine (even asymptotically) the values of md (k, n), but we are
nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if
an n-vertex k-uniform hypergraph G has minimum d-degree at least (1+γ )md (k, n)
(for any constantγ > 0), then the number of perfect matchings in G is controlled by
an entropy-like parameter of G. This strengthens cruder estimates arising from work
of Kang–Kelly–Kühn–Osthus–Pfenninger and Pham–Sah–Sawhney–Simkin.},
  author       = {Kwan, Matthew Alan and Safavi Hemami, Roodabeh and Wang, Yiting},
  issn         = {1439-6912},
  journal      = {Combinatorica},
  publisher    = {Springer Nature},
  title        = {{Counting perfect matchings in Dirac hypergraphs}},
  doi          = {10.1007/s00493-025-00194-8},
  volume       = {46},
  year         = {2026},
}

@article{15275,
  abstract     = {In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.},
  author       = {Fox, Jacob and Pach, János and Suk, Andrew},
  issn         = {1439-6912},
  journal      = {Combinatorica},
  keywords     = {Computational Mathematics, Discrete Mathematics and Combinatorics},
  number       = {6},
  pages        = {803--813},
  publisher    = {Springer Nature},
  title        = {{Bounded VC-dimension implies the Schur-Erdős conjecture}},
  doi          = {10.1007/s00493-021-4530-9},
  volume       = {41},
  year         = {2021},
}

@article{9582,
  abstract     = {The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least cH log d/log log d, thus nearly confirming one and proving another conjecture of Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdœs, Janson, Łuczak and Spencer.},
  author       = {Kwan, Matthew Alan and Letzter, Shoham and Sudakov, Benny and Tran, Tuan},
  issn         = {1439-6912},
  journal      = {Combinatorica},
  number       = {2},
  pages        = {283--305},
  publisher    = {Springer},
  title        = {{Dense induced bipartite subgraphs in triangle-free graphs}},
  doi          = {10.1007/s00493-019-4086-0},
  volume       = {40},
  year         = {2020},
}

@article{7034,
  abstract     = {We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus.},
  author       = {Fulek, Radoslav and Kynčl, Jan},
  issn         = {1439-6912},
  journal      = {Combinatorica},
  number       = {6},
  pages        = {1267--1279},
  publisher    = {Springer Nature},
  title        = {{Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4}},
  doi          = {10.1007/s00493-019-3905-7},
  volume       = {39},
  year         = {2019},
}

@article{4069,
  abstract     = {Let C be a cell complex in d-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope in d + 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in front/behind relation defined for the faces of C with respect to any fixed viewpoint x is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.},
  author       = {Edelsbrunner, Herbert},
  issn         = {1439-6912},
  journal      = {Combinatorica},
  number       = {3},
  pages        = {251 -- 260},
  publisher    = {Springer},
  title        = {{An acyclicity theorem for cell complexes in d dimension}},
  doi          = {10.1007/BF02122779},
  volume       = {10},
  year         = {1990},
}