@article{22168,
  abstract     = {Let us say that a graph G is Ramsey for a tuple (H1, ... , Hr) of graphs if every r-colouring
of the edges of G contains a monochromatic copy of Hi in colour i, for some i ∈ [[r]].
A famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and
Rucinski, predicts the threshold at which the binomial random graph ´ Gn,p becomes Ramsey
for (H1, ... , Hr) asymptotically almost surely.
In this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of
graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement,
which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs H1, ... , Hr by finite families
H1, ... , Hr. Additionally, we pose a natural (deterministic) graph-partitioning conjecture,
which we believe to be of independent interest, and whose resolution would imply the
Kohayakawa–Kreuter conjecture.},
  author       = {KUPERWASSER, EDEN and SAMOTIJ, WOJCIECH and Wigderson, Yuval},
  issn         = {1469-8064},
  journal      = {Mathematical Proceedings of the Cambridge Philosophical Society},
  number       = {3},
  pages        = {293--320},
  publisher    = {Cambridge University Press},
  title        = {{On the Kohayakawa–Kreuter conjecture}},
  doi          = {10.1017/s0305004125000143},
  volume       = {178},
  year         = {2025},
}

@article{9364,
  abstract     = {Let t : Fp → C be a complex valued function on Fp. A classical problem in analytic number theory is bounding the maximum M(t) := max 0≤H<p ∣ 1/√p ∑ 0≤n<H t (n) ∣ of the absolute value of the incomplete sums(1/√p)∑0≤n<H t (n). In this very general context one of the most important results is the Pólya–Vinogradov bound M(t)≤IIˆtII∞ log 3p, where ˆt : Fp → C is the normalized Fourier transform of t. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any ε > 0 there exists a large subset of a ∈ F×p such that for kl a,1,p : x → e((ax+x) / p) we have M(kla,1,p) ≥ (1−ε/√2π + o(1)) log log p, as p→∞. Finally, we prove a result on the growth of the moments of {M (kla,1,p)}a∈F×p. 2020 Mathematics Subject Classification: 11L03, 11T23 (Primary); 14F20, 60F10 (Secondary).},
  author       = {Bonolis, Dante},
  issn         = {1469-8064},
  journal      = {Mathematical Proceedings of the Cambridge Philosophical Society},
  number       = {3},
  pages        = {563 -- 590},
  publisher    = {Cambridge University Press},
  title        = {{On the size of the maximum of incomplete Kloosterman sums}},
  doi          = {10.1017/S030500412100030X},
  volume       = {172},
  year         = {2022},
}