@article{22153,
  abstract     = {A weakly optimal Ks-free (n,d,λ)-graph is a d-regular Ks-free graph on n vertices with d=Θ(n1−α) and spectral expansion λ=Θ(n1−(s−1)α), for some fixed α>0. Such a graph is called optimal if additionally α=12s−3. We prove that if s1,…,sk≥3 are fixed positive integers and weakly optimal Ksi-free pseudorandom graphs exist for each 1≤i≤k, then the multicolor Ramsey numbers satisfy
Ω(tS+1log2St)≤r(s1,…,sk,t)≤O(tS+1logSt),
as t→∞, where S=∑ki=1(si−2). This generalizes previous results of Mubayi and Verstraëte, who proved the case k=1, and Alon and Rödl, who proved the case s1=⋯=sk=3. Both previous results used the existence of optimal rather than weakly optimal Ksi-free graphs.},
  author       = {He, Xiaoyu and Wigderson, Yuval},
  issn         = {1077-8926},
  journal      = {The Electronic Journal of Combinatorics},
  number       = {1},
  publisher    = {The Electronic Journal of Combinatorics},
  title        = {{Multicolor Ramsey numbers via pseudorandom graphs}},
  doi          = {10.37236/9071},
  volume       = {27},
  year         = {2020},
}

@article{701,
  abstract     = {A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a^2, 3a^2 or a^2+b^2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m^3. We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m^2.},
  author       = {Kynčl, Jan and Patakova, Zuzana},
  issn         = {1077-8926},
  journal      = {The Electronic Journal of Combinatorics},
  number       = {3},
  pages        = {1--44},
  publisher    = {International Press of Boston},
  title        = {{On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4}},
  volume       = {24},
  year         = {2017},
}

@article{795,
  abstract     = {We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where every pair of independent edges crosses an even number of times, then G has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in D. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.},
  author       = {Fulek, Radoslav and Kynčl, Jan and Pálvölgyi, Dömötör},
  issn         = {1077-8926},
  journal      = {The Electronic Journal of Combinatorics},
  number       = {3},
  publisher    = {Electronic Journal of Combinatorics},
  title        = {{Unified Hanani Tutte theorem}},
  doi          = {10.37236/6663},
  volume       = {24},
  year         = {2017},
}