@article{22163,
  abstract     = {For a field F and integers d and k, a set A ⊆ Fd is called k-nearly orthogonal if its
members are non-self-orthogonal and every k + 1 vectors of A include an orthogonal pair.
We prove that for every prime p there exists some δ = δ(p)> 0, such that for every field
F of characteristic p and for all integers k ≥ 2 and d ≥ k, there exists a k-nearly orthogonal
set of at least dδ·k/ logk vectors of Fd. The size of the set is optimal up to the logk term
in the exponent. We further prove two extensions of this result. In the first, we provide a
large set A of non-self-orthogonal vectors of Fd such that for every two subsets of A of
size k+1 each, some vector of one of the subsets is orthogonal to some vector of the other.
In the second extension, every k + 1 vectors of the produced set A include ℓ + 1 pairwise
orthogonal vectors for an arbitrary fixed integer 1 ≤ ℓ ≤ k. The proofs involve probabilistic
and spectral arguments and the hypergraph container method},
  author       = {Haviv, Ishay and Mattheus, Sam and Milojević, Aleksa and Wigderson, Yuval},
  issn         = {0012-365X},
  journal      = {Discrete Mathematics},
  keywords     = {Nearly orthogonal sets, Ramsey theory, Finite fields},
  number       = {4},
  publisher    = {Elsevier},
  title        = {{Larger nearly orthogonal sets over finite fields}},
  doi          = {10.1016/j.disc.2024.114373},
  volume       = {348},
  year         = {2025},
}

@article{22165,
  abstract     = {The book graph 𝐵(𝑘)
𝑛 consists of 𝑛 copies of 𝐾𝑘+1 joined along a common 𝐾𝑘. In the prequel to this paper, we studied the diagonal Ramsey number 𝑟⁡(𝐵(𝑘)
𝑛,𝐵(𝑘)
𝑛). Here we consider the natural off-diagonal variant 𝑟⁡(𝐵(𝑘)
𝑐⁢𝑛,𝐵(𝑘)
𝑛) for fixed 𝑐 ∈(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.

},
  author       = {Conlon, David and Fox, Jacob and Wigderson, Yuval},
  issn         = {1469-2163},
  journal      = {Combinatorics, Probability and Computing},
  keywords     = {Ramsey theory, book graphs, Ramsey goodness},
  number       = {3},
  pages        = {516--545},
  publisher    = {Cambridge University Press},
  title        = {{Off-diagonal book Ramsey numbers}},
  doi          = {10.1017/s0963548322000360},
  volume       = {32},
  year         = {2023},
}