@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{22159, abstract = {The size Ramsey number of a graph H is defined as the minimum number of edges in a graph G such that there is a monochromatic copy of H in every two-coloring of E(G). The size Ramsey number was introduced by Erdős, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs (obtained by adding many common neighbors to the vertices of a clique), and starburst graphs (obtained by adding many pendant edges to each vertex of a clique). In this paper, we completely resolve the latter two questions and make substantial progress on the first by determining the size Ramsey number of Ks,t up to a constant factor for all t=Ω(s log s).}, author = {Conlon, David and Fox, Jacob and Wigderson, Yuval}, issn = {1439-6912}, journal = {Combinatorica}, number = {4}, pages = {743--768}, publisher = {Springer Nature}, title = {{Three early problems on size Ramsey numbers}}, doi = {10.1007/s00493-023-00034-7}, volume = {43}, year = {2023}, } @article{10335, abstract = {Van der Holst and Pendavingh introduced a graph parameter σ, which coincides with the more famous Colin de Verdière graph parameter μ for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on σ(G) which is, in general, tight. Equality between μ and σ does not hold in general as van der Holst and Pendavingh showed that there is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8. We also prove that, in general, the gap can be large: The incidence graphs Hq of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2.}, author = {Kaluza, Vojtech and Tancer, Martin}, issn = {0209-9683}, journal = {Combinatorica}, pages = {1317--1345}, publisher = {Springer Nature}, title = {{Even maps, the Colin de Verdière number and representations of graphs}}, doi = {10.1007/s00493-021-4443-7}, volume = {42}, year = {2022}, } @article{22174, abstract = {The book graph B (k) n consists of n copies of Kk+1 joined along a common Kk. The Ramsey numbers of B (k) n are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed k, thus answering an old question of Erd˝os, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemer´edi’s regularity lemma, thus providing much tighter control on the error term. Finally, we prove a conjecture of Nikiforov, Rousseau, and Schelp by showing that all extremal colorings for this Ramsey problem are quasirandom}, author = {Conlon, David and Fox, Jacob and Wigderson, Yuval}, issn = {1439-6912}, journal = {Combinatorica}, number = {3}, pages = {309--363}, publisher = {Springer Nature}, title = {{Ramsey numbers of books and quasirandomness}}, doi = {10.1007/s00493-021-4409-9}, volume = {42}, year = {2022}, } @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{1173, abstract = {We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions.}, author = {Edelsbrunner, Herbert and Glazyrin, Alexey and Musin, Oleg and Nikitenko, Anton}, issn = {0209-9683}, journal = {Combinatorica}, number = {5}, pages = {887 -- 910}, publisher = {Springer}, title = {{The Voronoi functional is maximized by the Delaunay triangulation in the plane}}, doi = {10.1007/s00493-016-3308-y}, volume = {37}, year = {2017}, } @article{4053, abstract = {We show that the maximum number of edges bounding m faces in an arrangement of n line segments in the plane is O(m2/3n2/3+nα(n)+nlog m). This improves a previous upper bound of Edelsbrunner et al. [5] and almost matches the best known lower bound which is Ω(m2/3n2/3+nα(n)). In addition, we show that the number of edges bounding any m faces in an arrangement of n line segments with a total of t intersecting pairs is O(m2/3t1/3+nα(t/n)+nmin{log m,log t/n}), almost matching the lower bound of Ω(m2/3t1/3+nα(t/n)) demonstrated in this paper.}, author = {Aronov, Boris and Edelsbrunner, Herbert and Guibas, Leonidas and Sharir, Micha}, issn = {0209-9683}, journal = {Combinatorica}, number = {3}, pages = {261 -- 274}, publisher = {Springer}, title = {{The number of edges of many faces in a line segment arrangement}}, doi = {10.1007/BF01285815}, volume = {12}, year = {1992}, } @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}, }