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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, Erdős, and Sós conjectured that the limit \begin{equation*} \lim _{n\to \infty } n^{-2} f^{(3)}(n;k+2,k) \end{equation*} exists for all k and confirmed it for k=2. Recently, Glock showed this for k=3. We settle the next open case, k=4, by showing that f^{(3)}(n;6,4)=\left (\frac {7}{36}+o(1)\right)n^2 as n\to \infty. More generally, for all k\in \{3,4\}, r\ge 3 and t\in [2,r-1], we compute the value of the limit \lim _{n\to \infty } n^{-t}f^{(r)}(n;k(r-t)+t,k), which settles a problem of Shangguan and Tamo. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular, uniform hypergraph H$\mathcal {H}$ with small maximum codegree has an almost‐perfect matching. We extend this result by obtaining a conflict‐free matching, where conflicts are encoded via a collection C$\mathcal {C}$ of subsets C⊆E(H)$C\subseteq E(\mathcal {H})$. We say that a matching M⊆E(H)$\mathcal {M}\subseteq E(\mathcal {H})$ is conflict‐free if M$\mathcal {M}$ does not contain an element of C$\mathcal {C}$ as a subset. Under natural assumptions on C$\mathcal {C}$, we prove that H$\mathcal {H}$ has a conflict‐free, almost‐perfect matching. This has many applications, one of which yields new asymptotic results for so‐called 'high‐girth' Steiner systems. Our main tool is a random greedy algorithm which we call the 'conflict‐free matching process'. [ABSTRACT FROM AUTHOR]

We prove that for any integer k≥2$$ k\ge 2 $$ and ε>0$$ \varepsilon >0 $$, there is an integer ℓ0≥1$$ {\ell}_0\ge 1 $$ such that any k‐uniform hypergraph on n vertices with minimum codegree at least (1/2+ε)n$$ \left(1/2+\varepsilon \right)n $$ has a fractional decomposition into (tight) cycles of length ℓ$$ \ell $$ (ℓ$$ \ell $$‐cycles for short) whenever ℓ≥ℓ0$$ \ell \ge {\ell}_0 $$ and n is large in terms of ℓ$$ \ell $$. This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into ℓ$$ \ell $$‐cycles. Moreover, for graphs this even guarantees integral decompositions into ℓ$$ \ell $$‐cycles and solves a problem posed by Glock, Kühn, and Osthus. For our proof, we introduce a new method for finding a set of ℓ$$ \ell $$‐cycles such that every edge is contained in roughly the same number of ℓ$$ \ell $$‐cycles from this set by exploiting that certain Markov chains are rapidly mixing. [ABSTRACT FROM AUTHOR]

Peripheral neuropathy is a common complication of arsenic toxicity. Symptoms are usually mild and reversible following discontinuation of treatment. A more severe chronic sensorimotor polyneuropathy characterized by distal axonal-loss neuropathy can be seen in chronic arsenic exposure. The clinical course of arsenic neurotoxicity in patients with coexistence of thiamine deficiency is only anecdotally known but this association may potentially lead to severe consequences. We describe a case of acute irreversible axonal neuropathy in a patient with hidden thiamine deficiency who was treated with a short course of arsenic trioxide for acute promyelocytic leukemia. Thiamine replacement therapy and arsenic trioxide discontinuation were not followed by neurological recovery and severe polyneuropathy persisted at 12-month follow-up. Thiamine plasma levels should be measured in patients who are candidate to arsenic trioxide therapy. Prophylactic administration of vitamin B1 may be advisable. The appearance of polyneuropathy signs early during the administration of arsenic trioxide should prompt electrodiagnostic testing to rule out a pattern of axonal neuropathy which would need immediate discontinuation of arsenic trioxide. [ABSTRACT FROM AUTHOR]