On upper transversals in 3-uniform hypergraphs
Summary: A set \(S\) of vertices in a hypergraph \(H\) is a transversal if it has a nonempty intersection with every edge of \(H\). The upper transversal number \(\Upsilon(H)\) of \(H\) is the maximum cardinality of a minimal transversal in \(H\). We show that if \(H\) is a connected \(3\)-uniform hypergraph of order \(n\), then \(\Upsilon(H) > 1.4855 \root{3}\of{n} - 2\). For \(n\) sufficiently large, we construct infinitely many connected \(3\)-uniform hypergraphs, \(H\), of order \(n\) satisfying \(\Upsilon(H) < 2.5199 \root{3}\of{n}\). We conjecture that \(\sup_{n\rightarrow\infty} \bigg(\text{inf}\frac{\Upsilon(H)}{\root{3}\of{n}}\bigg) = \root{3}\of{16}\), where the infimum is taken over all connected \(3\)-uniform hypergraphs \(H\) of order \(n\).
scientific article; zbMATH DE number 6983025