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7008 results in Algorithmics, Complexity, Computer Algebra, Computational Geometry

Page 3 of 351

  • 1 - Introduction

  • Joseph O'Rourke, Smith College, Massachusetts
  • Book:
    The Mathematics of Origami
    Published online:
    28 November 2025
    Print publication:
    18 December 2025, pp 1-5
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  • Hypergraphs with uniform Turán density equal to 8/27

  • Part of
  • Frederik Garbe, Daniel Il’kovič, Daniel Kráľ, Filip Kučerák, Ander Lamaison
  • Journal:
    Combinatorics, Probability and Computing / Volume 35 / Issue 2 / March 2026
    Published online by Cambridge University Press:
    17 December 2025, pp. 280-289
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  • In the 1980s, Erdős and Sós initiated the study of Turán problems with a uniformity condition on the distribution of edges: the uniform Turán density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least $d$ contains $H$. In particular, they asked to determine the uniform Turán densities of $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349 – 366] and [J. Eur. Math. Soc. 20 (2018), 1139 – 1159], while the latter still remains open. Till today, there are known constructions of $3$-uniform hypergraphs with uniform Turán density equal to $0$, $1/27$, $4/27$, and $1/4$ only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Turán density to be equal to $8/27$ and identify hypergraphs satisfying this condition.

  • Hypergraphs without complete partite subgraphs

  • Part of
  • Dhruv Mubayi
  • Journal:
    Combinatorics, Probability and Computing , First View
    Published online by Cambridge University Press:
    11 December 2025, pp. 1-4
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  • Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod _{i=1}^{r-1}s_i$. Let $K=K(s_1, \ldots , s_{r-1}, t)$ denote the complete $r$-partite $r$-uniform hypergraph with parts of size $s_1, \ldots , s_{r-1}, t$. We prove that the Zarankiewicz number $z(n, K)= n^{r-1/s-o(1)}$ provided $t\gt 3^{s+o(s)}$. Previously this was known only for $t \gt ((r-1)(s-1))!$ due to Pohoata and Zakharov. Our novel approach, which uses Behrend’s construction of sets with no 3-term arithmetic progression, also applies for small values of $s_i$, for example, it gives $z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.

  • Sidorenko’s conjecture for subdivisions and theta substitutions

  • Part of
  • Seonghyuk Im, Ruonan Li, Hong Liu
  • Journal:
    Combinatorics, Probability and Computing / Volume 35 / Issue 2 / March 2026
    Published online by Cambridge University Press:
    10 December 2025, pp. 269-279
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  • The famous Sidorenko’s conjecture asserts that for every bipartite graph $H$, the number of homomorphisms from $H$ to a graph $G$ with given edge density is minimised when $G$ is pseudorandom. We prove that for any graph $H$, a graph obtained from replacing edges of $H$ by generalised theta graphs consisting of even paths satisfies Sidorenko’s conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalised theta graph satisfy Sidorenko’s conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].

Page 3 of 351