skip to main content

Search for: All records

Creators/Authors contains: "Conlon, David"

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. Free, publicly-accessible full text available June 1, 2025
  2. Free, publicly-accessible full text available January 1, 2025
  3. Free, publicly-accessible full text available December 1, 2024
  4. Free, publicly-accessible full text available January 1, 2025
  5. Abstract

    We show that the size-Ramsey number of the$\sqrt{n} \times \sqrt{n}$grid graph is$O(n^{5/4})$, improving a previous bound of$n^{3/2 + o(1)}$by Clemens, Miralaei, Reding, Schacht, and Taraz.

    more » « less
    Free, publicly-accessible full text available November 1, 2024
  6. Abstract

    Let denote the complete 3‐uniform hypergraph on vertices and the 3‐uniform hypergraph on vertices consisting of all edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off‐diagonal Ramsey number exhibits an unusual intermediate growth rate, namely,for some positive constants and . The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum such that any 2‐edge‐coloring of the Cartesian product contains either a red rectangle or a blue ?

    more » « less
    Free, publicly-accessible full text available October 1, 2024
  7. Free, publicly-accessible full text available August 1, 2024
  8. Abstract

    Building on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form for a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an by ‐matrix that does not have any matrix from a fixed finite family derived from as a submatrix. As an application, we give new lower bounds for the Ramsey numbers and , namely, and . We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of for any fixed integer .

    more » « less
  9. Abstract The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $c$ , a simple $k$ -partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being $k$ -partite, while, for $c$ bounded away from $0$ , random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$ . 
    more » « less
  10. Abstract

    We show that there is an absolute constant such that for any finite subset of and any transcendental number . By a construction of Konyagin and Łaba, this is best possible up to the constant .

    more » « less