Skip to main content Accessibility help
×
  1. Home
  2. Browse subjects
  3. Mathematics
  4. Discrete Mathematics, Information Theory and Coding
  5. Content listing

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

9084 results in Discrete Mathematics, Information Theory and Coding

Page 3 of 455

  • Clique density vs blowups

  • Part of
  • Domagoj Bradac, Hong Liu, Zhuo Wu, Zixiang Xu
  • Journal:
    Combinatorics, Probability and Computing , First View
    Published online by Cambridge University Press:
    22 January 2026, pp. 1-22
    • Article
      • You have access
      • Open access
    • PDF
    • HTML
    • Export citation

  • A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov’s result in the following form. Given $r,t\in \mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include:

    • • For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline {G}$ has positive triangle density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline {G}$ has positive $K_r$-density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2.

    • • Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $\Omega ({n \over {\log n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $\Omega (n)$, where the constant $(2h-2)^{r}+1$ is optimal.

    The ${n \over {\log n}}$ size of the blowups in all our results are optimal up to a constant factor.

Convex Polytopes and Polyhedra
  • Peter McMullen
  • Published online:
    22 December 2025
    Print publication:
    22 January 2026
  • A valuable resource for researchers in discrete and combinatorial geometry, this book offers comprehensive coverage of several modern developments on algebraic and combinatorial properties of polytopes. The introductory chapters provide a new approach to the basic properties of convex polyhedra and how they are connected; for instance, fibre operations are treated early on. Finite tilings and polyhedral convex functions play an important role, and lead to the new technique of tiling diagrams. Special classes of polytopes such as zonotopes also have corresponding diagrams. A central result is the complete characterization of the possible face-numbers of simple polytopes. Tools used for this are representations and the weight algebra of mixed volumes. An unexpected consequence of the proof is an algebraic treatment of Brunn–Minkowski theory as applied to polytopes. Valuations also provide a thread running through the book, and the abstract theory and related tensor algebras are treated in detail.

  • 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

  • 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.

Page 3 of 455