78 research outputs found

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

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

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

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

### Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings

In 1973, Erd\H{o}s conjectured the existence of high girth $(n,3,2)$-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 $(n,q,r)$-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 $G$ avoiding a given set of
submatchings (which we view as a hypergraph $H$ where $V(H)=E(G)$). 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 $H$ 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 $K_n^r$ into approximate high
girth $(n,q,r)$-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 $\Delta$
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

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is
$(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason
independently proved that every graph with no $K_t$ minor has average degree
$O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently,
Norin, Song and the second author showed that every graph with no $K_t$ minor
is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1/4$, making the first
improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. The
first main result of this paper is that every graph with no $K_t$ minor is
$O(t\log\log t)$-colorable.
This is a corollary of our main technical result that the chromatic number of
a $K_t$-minor-free graph is bounded by $O(t(1+f(G,t)))$ where $f(G,t)$ is the
maximum of $\frac{\chi(H)}{a}$ over all $a\ge \frac{t}{\sqrt{\log t}}$ and
$K_a$-minor-free subgraphs $H$ of $G$ that are small (i.e. $O(a\log^4 a)$
vertices). This has a number of interesting corollaries. First as mentioned,
using the current best-known bounds on coloring small $K_t$-minor-free graphs,
we show that $K_t$-minor-free graphs are $O(t\log\log t)$-colorable. Second, it
shows that proving Linear Hadwiger's Conjecture (that $K_t$-minor-free graphs
are $O(t)$-colorable) reduces to proving it for small graphs. Third, we prove
that $K_t$-minor-free graphs with clique number at most $\sqrt{\log t}/ (\log
\log t)^2$ are $O(t)$-colorable. This implies our final corollary that Linear
Hadwiger's Conjecture holds for $K_r$-free graphs for every fixed $r$.
One key to proving the main theorem is a new standalone result that every
$K_t$-minor-free graph of average degree $d=\Omega(t)$ has a subgraph on $O(t
\log^3 t)$ vertices with average degree $\Omega(d)$.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

A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ such that every
$p$-clique receives at least $q$ colors. In 1975, Erd\H{o}s and Shelah
introduced the generalized Ramsey number $f(n,p,q)$ which is the minimum number
of colors needed in a $(p,q)$-coloring of $K_n$. In 1997, Erd\H{o}s and
Gy\'arf\'as showed that $f(n,p,q)$ is at most a constant times
$n^{\frac{p-2}{\binom{p}{2} - q + 1}}$. Very recently the first author, Dudek,
and English improved this bound by a factor of $\log n^{\frac{-1}{\binom{p}{2}
- q + 1}}$ for all $q \le \frac{p^2 - 26p + 55}{4}$, and they ask if this
improvement could hold for a wider range of $q$.
We answer this in the affirmative for the entire non-integral regime, that
is, for all integers $p, q$ with $p-2$ not divisible by $\binom{p}{2} - q + 1$.
Furthermore, we provide a simultaneous three-way generalization as follows:
where $p$-clique is replaced by any fixed graph $F$ (with $|V(F)|-2$ not
divisible by $|E(F)| - q + 1$); to list coloring; and to $k$-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

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

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

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