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- W3209968583 abstract "Aharoni and Howard and, independently, Huang et al. (2012) proposed the following rainbow version of the Erdős matching conjecture: For positive integers n, k and m with n ⩾ km, if each of the families $$F_{1},ldots,F_{m}subseteqleft(begin{array}{c}[n] kend{array}right)$$ has size more than $$max{left(begin{array}{c}n kend{array}right)-left(begin{array}{c}n-m+1 kend{array}right),left(begin{array}{c}km-1 kend{array}right)}$$ , then there exist pairwise disjoint subsets e1,…,em such that ei ∈ Fi for all i ∈ [m]. We prove that there exists an absolute constant n0 such that this rainbow version holds for k = 3 and n ⩾ n0. We convert this rainbow matching problem to a matching problem on a special hypergraph H. We then combine several existing techniques on matchings in uniform hypergraphs: Find an absorbing matching M in H; use a randomization process of Alon et al. (2012) to find an almost regular subgraph of H − V(M); find an almost perfect matching in H − V(M). To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of Łuczak and Mieczkowska (2014) and might be of independent interest." @default.
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- W3209968583 date "2021-10-29" @default.
- W3209968583 modified "2023-10-17" @default.
- W3209968583 title "On the rainbow matching conjecture for 3-uniform hypergraphs" @default.
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- W3209968583 doi "https://doi.org/10.1007/s11425-020-1890-4" @default.
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