Independence in Uniform Linear Triangle-free Hypergraphs

The independence number $\alpha (H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer’s classical lower bound on the independence number of triangle-free graphs (J. Comb. Theory, Ser. B 53 (1991) 300-307), and considerably improving recent results of Li and Zang (SIAM J. Discrete Math. 20 (2006) 96-104) and Chishti et al. (Acta Univ. Sapientiae, Informatica 6 (2014) 132-158), we show that $$\alpha (H)\ge \sum _{u\in V(H)}{f}_r(d_H(u))$$ for an $r$-uniform linear triangle-free hypergraph $H$ with $r\ge 2$, where \begin{eqnarray*} f_r(0)&=&1\mbox{, and }\ f_r(d)&=&\frac{1+\Big((r-1)d^2-d\Big)f_r(d-1)}{1+(r-1)d^2}\mbox{ for $d\geq 1$.} \end{eqnarray*}