11 research outputs found

    On the maximum size of connected hypergraphs without a path of given length

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    In this note we asymptotically determine the maximum number of hyperedges possible in an rr-uniform, connected nn-vertex hypergraph without a Berge path of length kk, as nn and kk tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity

    Stability of extremal connected hypergraphs avoiding Berge-paths

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    A Berge-path of length kk in a hypergraph H\mathcal{H} is a sequence v1,e1,v2,e2,…,vk,ek,vk+1v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1} of distinct vertices and hyperedges with vi+1∈ei,ei+1v_{i+1}\in e_i,e_{i+1} for all i∈[k]i\in[k]. F\"uredi, Kostochka and Luo, and independently Gy\H{o}ri, Salia and Zamora determined the maximum number of hyperedges in an nn-vertex, connected, rr-uniform hypergraph that does not contain a Berge-path of length kk provided kk is large enough compared to rr. They also determined the unique extremal hypergraph H1\mathcal{H}_1. We prove a stability version of this result by presenting another construction H2\mathcal{H}_2 and showing that any nn-vertex, connected, rr-uniform hypergraph without a Berge-path of length kk, that contains more than ∣H2∣|\mathcal{H}_2| hyperedges must be a subhypergraph of the extremal hypergraph H1\mathcal{H}_1, provided kk is large enough compared to rr
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