@article{11740,
  abstract     = {We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each k, each set of k+1 vertices forms an edge with some probability pk independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408–417].
We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group R. We prove that this notion of connectedness displays a phase transition and determine the threshold. We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition.},
  author       = {Cooley, Oliver and Del Giudice, Nicola and Kang, Mihyun and Sprüssel, Philipp},
  issn         = {1077-8926},
  journal      = {Electronic Journal of Combinatorics},
  number       = {3},
  publisher    = {Electronic Journal of Combinatorics},
  title        = {{Phase transition in cohomology groups of non-uniform random simplicial complexes}},
  doi          = {10.37236/10607},
  volume       = {29},
  year         = {2022},
}

@article{12151,
  abstract     = {The k-sample G(k,W) from a graphon W:[0,1]2→[0,1] is the random graph on {1,…,k}, where we sample x1,…,xk∈[0,1] uniformly at random and make each pair {i,j}⊆{1,…,k} an edge with probability W(xi,xj), with all these choices being mutually independent. Let the random variable Xk(W) be the number of edges in  G(k,W). Vera T. Sós asked in 2012 whether two graphons U, W are necessarily weakly isomorphic if the random variables Xk(U) and Xk(W) have the same distribution for every integer k≥2. This question when one of the graphons W is a constant function was answered positively by Endre Csóka and independently by Jacob Fox, Tomasz Łuczak and Vera T. Sós. Here we investigate the question when W is a 2-step graphon and prove that the answer is positive for a 3-dimensional family of such graphons. We also present some related results.},
  author       = {Cooley, Oliver and Kang, M. and Pikhurko, O.},
  issn         = {1588-2632},
  journal      = {Acta Mathematica Hungarica},
  keywords     = {graphon, k-sample, graphon forcing, graph container},
  pages        = {1--26},
  publisher    = {Springer Nature},
  title        = {{On a question of Vera T. Sós about size forcing of graphons}},
  doi          = {10.1007/s10474-022-01265-8},
  volume       = {168},
  year         = {2022},
}

@article{12286,
  abstract     = {Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes.
Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in $H^r(n,p)$.},
  author       = {Cooley, Oliver and Kang, Mihyun and Zalla, Julian},
  issn         = {1077-8926},
  journal      = {The Electronic Journal of Combinatorics},
  keywords     = {Computational Theory and Mathematics, Geometry and Topology, Theoretical Computer Science, Applied Mathematics, Discrete Mathematics and Combinatorics},
  number       = {4},
  publisher    = {The Electronic Journal of Combinatorics},
  title        = {{Loose cores and cycles in random hypergraphs}},
  doi          = {10.37236/10794},
  volume       = {29},
  year         = {2022},
}