Bounds on upper transversals in hypergraphs

A set S of vertices in a hypergraph H is a transversal if it has a nonempty intersection with every edge of H. For $$k \ge 1$$k≥1, if H is a hypergraph with every edge of size at least k, then a k-transversal in H is a transversal that intersects every edge of H in at least k vertices. In particular, a 1-transversal is a transversal. The upper k-transversal number $$\Upsilon _{k}(H)$$Υk(H) of H is the maximum cardinality of a minimal k-transversal in H. Let H be a hypergraph with $$n_{_H}$$nH vertices and $$m_{_H}$$mH edges. We show that for $$r \ge 2$$r≥2 and for every integer $$k \in [r]$$k∈[r], if H is r-uniform with maximum degree $$\Delta $$Δ, then $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{k (\Delta - 1) + r} \right) n_{_H}$$Υk(H)≤k·Δk(Δ-1)+rnH and $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{\Delta (k + 1) + r - k} \right) (n_{_H}+ m_{_H})$$Υk(H)≤k·ΔΔ(k+1)+r-k(nH+mH), and both bounds are tight. As a special case of this result, if H is a 3-regular, 3-uniform hypergraph, then $$\Upsilon _{2}(H) \le \frac{6}{7} n_{_H}$$Υ2(H)≤67nH, and equality in this bound is achieved by the Fano plane. We also discuss a relation between upper transversals in 3-uniform hypergraphs and the famous cap set problem, and show that for every given $$\epsilon > 0$$ϵ>0, there exists a 3-uniform, connected, linear hypergraphs of sufficiently large order such that $$\Upsilon _{1}(H)