We prove that for any integer k ≥ 2 and 𝜀 > 0, there is an integer 𝓁 0 ≥ 1 such that any k-uniform hypergraph on n vertices with minimum codegree at least (1∕2 + 𝜀)n has a fractional decomposition into (tight) cycles of length 𝓁 (𝓁-cycles for short) whenever 𝓁 ≥ 𝓁 0 and n is large in terms of 𝓁. This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into 𝓁-cycles. Moreover, for graphs this even guarantees integral decompositions into 𝓁-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-cycles such that every edge is contained in roughly the same number of 𝓁-cycles from this set by exploiting that certain Markov chains are rapidly mixing.

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.

We prove that for any integer k ≥ 2 and ε > 0, there is an integer ℓ0 ≥ 1 such that any k-uniform hypergraph on n vertices with minimum codegree at least (1/2 + ε)n has a fractional decomposition into tight cycles of length ℓ (ℓ-cycles for short) whenever ℓ ≥ ℓ0 and n is large in terms of ℓ. This is essentially tight.This immediately yields also approximate integral decompositions for these hypergraphs into ℓ-cycles. Moreover, for graphs this even guarantees integral decompositions into ℓ-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-cycles such that every edge is contained in roughly the same number of ℓ-cycles from this set by exploiting that certain Markov chains are rapidly mixing.The research leading to these results was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -428212407. − − −⇀ f are adjacent if there is an ordering of k + 1 vertices of H such that − −⇀ e coincides with the first k vertices and − − −⇀ f with the last k vertices and our restrictions allow to transition from − −⇀ e to − − −⇀ f .

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almostregular, uniform hypergraph 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 of subsets C ⊆ E(H). We say that a matching M ⊆ E(H) is conflict-free if M does not contain an element of C as a subset. Under natural assumptions on C, we prove that 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".

A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in k-uniform hypergraphs H on n vertices with minimum pk ´1q-degree δ k´1 pHq ě p1{2 `op1qqn, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on n vertices with δpGq ě p1{2 `op1qqn contains p1 ´op1qqr edgedisjoint Hamilton cycles where r is the largest integer such that G contains a spanning 2r-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised by Kühn, Lapinskas, and Osthus.We extend this result to hypergraphs; every k-uniform hypergraph H on n vertices with δ k´1 pHq ě p1{2 `op1qqn contains p1 ´op1qqr edge-disjoint (tight) Hamilton cycles where r is the largest integer such that H contains a spanning subgraph with each vertex belonging to kr edges. In particular, this yields an asymptotic solution to a question of Glock, Kühn, and Osthus.In fact, our main result applies to approximately vertex-regular k-uniform hypergraphs with a weak quasirandom property and provides approximate decompositions into cycle factors without too short cycles.