On Subgraphs of Bounded Degeneracy in Hypergraphs

A $k$-uniform hypergraph has degeneracy bounded by $d$ if every induced subgraph has a vertex of degree at most $d$. Given a $k$-uniform hypergraph $H = (V (H), E(H))$, we show there exists an induced subgraph of size at least $\sum{v\in V (H)} \min \left\{1, c_k\left(\frac{ d + 1}{ d_{H (v)} + 1}\right)^{1/(k−1)}\right\}$, where $c_k = 2^{−\left(1+ \frac{1}{ k−1}\right)}(1-1/k)$ and $d_{H (v)}$ denotes the degree of vertex $v$ in the hypergraph $H$. This extends and generalizes a result of Alon-Kahn-Seymour (Graphs and Combinatorics, 1987) for graphs, as well as a result of Dutta-Mubayi-Subramanian (SIAM Journal on Discrete Mathematics, 2012) for linear hypergraphs, to general $k$-uniform hypergraphs. We also generalize the results of Srinivasan and Shachnai (SIAM Journal on Discrete Mathematics, 2004) from independent sets (0-degenerate subgraphs) to d-degenerate subgraphs. We further give a simple non-probabilistic proof of the Dutta-Mubayi-Subramanian bound for linear k-uniform hypergraphs, which extends the Alon-Kahn-Seymour (Graphs and Combinatorics, 1987) proof technique to hypergraphs. Our proof combines the random permutation technique of Bopanna-Caro-Wei (see e.g. The Probabilistic Method, N. Alon and J. H. Spencer; Dutta-Mubayi-Subramanian) and also Beame-Luby (SODA, 1990) together with a new local density argument which may be of independent interest. Our results also imply some results in discrete geometry, and we further address some natural algorithmic questions.