78 research outputs found

    Alveolar hyperoxia and exacerbation of lung injury in critically Ill SARS-CoV-2 pneumonia

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    Acute hypoxic respiratory failure (AHRF) is a prominent feature of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) critical illness. The severity of gas exchange impairment correlates with worse prognosis, and AHRF requiring mechanical ventilation is associated with substantial mortality. Persistent impaired gas exchange leading to hypoxemia often warrants the prolonged administration of a high fraction of inspired oxygen (FiO2). In SARS-CoV-2 AHRF, systemic vasculopathy with lung microthrombosis and microangiopathy further exacerbates poor gas exchange due to alveolar inflammation and oedema. Capillary congestion with microthrombosis is a common autopsy finding in the lungs of patients who die with coronavirus disease 2019 (COVID-19)-associated acute respiratory distress syndrome. The need for a high FiO2 to normalise arterial hypoxemia and tissue hypoxia can result in alveolar hyperoxia. This in turn can lead to local alveolar oxidative stress with associated inflammation, alveolar epithelial cell apoptosis, surfactant dysfunction, pulmonary vascular abnormalities, resorption atelectasis, and impairment of innate immunity predisposing to secondary bacterial infections. While oxygen is a life-saving treatment, alveolar hyperoxia may exacerbate pre-existing lung injury. In this review, we provide a summary of oxygen toxicity mechanisms, evaluating the consequences of alveolar hyperoxia in COVID-19 and propose established and potential exploratory treatment pathways to minimise alveolar hyperoxia.</p

    Decomposing random regular graphs into stars

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    We study k-star decompositions, that is, partitions of the edge set into disjoint stars with k edges, in the uniformly random d-regular graph model Gn,d. We prove an existence result for such decompositions for all d,k such that d/2d/2, and strongly suggest the a.a.s. existence of such decompositions is equivalent to the a.a.s. existence of independent sets of size (2k−d)n/(2k), subject to the necessary divisibility conditions on the number of vertices. For smaller values of k, the connection between k-star decompositions and β-orientations allows us to apply results of Thomassen (2012) and Lovász, Thomassen, Wu and Zhang (2013). We prove that random d-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of k-star decompositions (i) when 2k2+k≤d, and (ii) when k is odd and k<d/2.This preprint is available through arXiv at doi:https://doi.org/10.48550/arXiv.2308.16037. Copyright 2023, The Authors. Posted with permission

    The limit in the (k+2,k)(k+2, k)-Problem of Brown, Erd\H{o}s and S\'os exists for all k2k\geq 2

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    Let f(r)(n;s,k)f^{(r)}(n;s,k) be the maximum number of edges of an rr-uniform hypergraph on~nn vertices not containing a subgraph with kk~edges and at most ss~vertices. In 1973, Brown, Erd\H{o}s and S\'os conjectured that the limit limnn2f(3)(n;k+2,k)\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k) exists for all positive integers k2k\ge 2. They proved this for k=2k=2. In 2019, Glock proved this for k=3k=3 and determined the limit. Quite recently, Glock, Joos, Kim, K\"{u}hn, Lichev and Pikhurko proved this for k=4k=4 and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers k2k\ge 2.Comment: 10 pages, to appear in Proceedings of the AM

    Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings

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    In 1973, Erd\H{o}s conjectured the existence of high girth (n,3,2)(n,3,2)-Steiner systems. Recently, Glock, K\"{u}hn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erd\H{o}s' conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erd\H{o}s' conjecture. As for Steiner systems with more general parameters, Glock, K\"{u}hn, Lo, and Osthus conjectured the existence of high girth (n,q,r)(n,q,r)-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph GG avoiding a given set of submatchings (which we view as a hypergraph HH where V(H)=E(G)V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Koml\'os, Pintz, Spencer, and Szemer\'edi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when HH is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed, the coloring version of our result even yields an almost partition of KnrK_n^r into approximate high girth (n,q,r)(n,q,r)-Steiner systems. Our main results also imply the existence of a perfect matching in a bipartite hypergraph where the parts have slightly unbalanced degrees. This has a number of applications; for example, it proves the existence of Δ\Delta pairwise disjoint list colorings in the setting of Kahn's theorem; it also proves asymptotic versions of various rainbow matching results in the sparse setting (where the number of times a color appears could be much smaller than the number of colors) and even the existence of many pairwise disjoint rainbow matchings in such circumstances.Comment: 52 page

    Reducing Linear Hadwiger's Conjecture to Coloring Small Graphs

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    In 1943, Hadwiger conjectured that every graph with no KtK_t minor is (t1)(t-1)-colorable for every t1t\ge 1. In the 1980s, Kostochka and Thomason independently proved that every graph with no KtK_t minor has average degree O(tlogt)O(t\sqrt{\log t}) and hence is O(tlogt)O(t\sqrt{\log t})-colorable. Recently, Norin, Song and the second author showed that every graph with no KtK_t minor is O(t(logt)β)O(t(\log t)^{\beta})-colorable for every β>1/4\beta > 1/4, making the first improvement on the order of magnitude of the O(tlogt)O(t\sqrt{\log t}) bound. The first main result of this paper is that every graph with no KtK_t minor is O(tloglogt)O(t\log\log t)-colorable. This is a corollary of our main technical result that the chromatic number of a KtK_t-minor-free graph is bounded by O(t(1+f(G,t)))O(t(1+f(G,t))) where f(G,t)f(G,t) is the maximum of χ(H)a\frac{\chi(H)}{a} over all atlogta\ge \frac{t}{\sqrt{\log t}} and KaK_a-minor-free subgraphs HH of GG that are small (i.e. O(alog4a)O(a\log^4 a) vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small KtK_t-minor-free graphs, we show that KtK_t-minor-free graphs are O(tloglogt)O(t\log\log t)-colorable. Second, it shows that proving Linear Hadwiger's Conjecture (that KtK_t-minor-free graphs are O(t)O(t)-colorable) reduces to proving it for small graphs. Third, we prove that KtK_t-minor-free graphs with clique number at most logt/(loglogt)2\sqrt{\log t}/ (\log \log t)^2 are O(t)O(t)-colorable. This implies our final corollary that Linear Hadwiger's Conjecture holds for KrK_r-free graphs for every fixed rr. One key to proving the main theorem is a new standalone result that every KtK_t-minor-free graph of average degree d=Ω(t)d=\Omega(t) has a subgraph on O(tlog3t)O(t \log^3 t) vertices with average degree Ω(d)\Omega(d).Comment: 24 pages. This and previous version add the necessary results from arXiv:2006.11798 in order to create a self-contained standalone paper. arXiv admin note: text overlap with arXiv:2006.11798, arXiv:2010.0599

    On generalized Ramsey numbers in the non-integral regime

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    A (p,q)(p,q)-coloring of a graph GG is an edge-coloring of GG such that every pp-clique receives at least qq colors. In 1975, Erd\H{o}s and Shelah introduced the generalized Ramsey number f(n,p,q)f(n,p,q) which is the minimum number of colors needed in a (p,q)(p,q)-coloring of KnK_n. In 1997, Erd\H{o}s and Gy\'arf\'as showed that f(n,p,q)f(n,p,q) is at most a constant times np2(p2)q+1n^{\frac{p-2}{\binom{p}{2} - q + 1}}. Very recently the first author, Dudek, and English improved this bound by a factor of logn1(p2)q+1\log n^{\frac{-1}{\binom{p}{2} - q + 1}} for all qp226p+554q \le \frac{p^2 - 26p + 55}{4}, and they ask if this improvement could hold for a wider range of qq. We answer this in the affirmative for the entire non-integral regime, that is, for all integers p,qp, q with p2p-2 not divisible by (p2)q+1\binom{p}{2} - q + 1. Furthermore, we provide a simultaneous three-way generalization as follows: where pp-clique is replaced by any fixed graph FF (with V(F)2|V(F)|-2 not divisible by E(F)q+1|E(F)| - q + 1); to list coloring; and to kk-uniform hypergraphs. Our results are a new application of the Forbidden Submatching Method of the second and fourth authors.Comment: 9 pages; new version extends results from sublinear regime to entire non-integral regime; new co-author adde

    11/4-colorability of subcubic triangle-free graphs

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    We prove that up to two exceptions, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4. This is tight unless further exceptional graphs are excluded, and improves the known bound on the fractional chromatic number of subcubic triangle-free planar graphs.This preprint is made available through arXiv at doi:https://doi.org/10.48550/arXiv.2204.12683. This work is licensed under the Creative Commons Attribution 4.0 License

    Local Hadwiger's Conjecture

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    We propose local versions of Hadwiger's Conjecture, where only balls of radius Ω(logv(G))\Omega(\log v(G)) around each vertex are required to be KtK_{t}-minor-free. We ask: if a graph is locally-KtK_{t}-minor-free, is it tt-colourable? We show that the answer is yes when t5t \leq 5, even in the stronger setting of list-colouring, and we complement this result with a O(logv(G))O(\log v(G))-round distributed colouring algorithm in the LOCAL model. Further, we show that for large enough values of tt, we can list-colour locally-KtK_{t}-minor-free graphs with 13max{h(t),312(t1)}13 \cdot \max\left\{h(t),\left\lceil \frac{31}{2}(t-1) \right\rceil \right\} colours, where h(t)h(t) is any value such that all KtK_{t}-minor-free graphs are h(t)h(t)-list-colourable. We again complement this with a O(logv(G))O(\log v(G))-round distributed algorithm.Comment: 24 pages; some minor typos have been fixe