3-UNIFORM HYPERGRAPHS AND LINEAR CYCLES.

Gyárfás, Győri, and Simonovits [J. Comb., 7 (2016), pp. 205{216] proved that if a 3-uniform hypergraph with n vertices has no linear cycles, then its independence number α ≥ 2n/5. The hypergraph consisting of vertex disjoint copies of a complete hypergraph K3/5 on five vertices shows that equality can hold. They asked whether this bound can be improved if we exclude K3/5 as a subhypergraph and whether such a hypergraph is 2-colorable. In this paper, we answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph doesn't contain K3 5 as a subhypergraph, then it is 2-colorable. This result clearly implies that its independence number α ≥ [n/2]. We show that this bound is sharp. Gyárfás, Gy}ori, and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n - 2 when n ≥ 10. [ABSTRACT FROM AUTHOR]