A note on the transversal size of a series of families constructed over Cycle Graph

TL;DR

This paper investigates the transversal size of specific intersecting families built over cycle graphs, providing an example with a number of blocks exceeding previous known bounds, thus contributing to extremal combinatorics.

Contribution

It introduces a new series of uniform intersecting families over cycle graphs with more blocks than previously known, challenging existing conjectures.

Findings

Constructed a series of families with over $(k/2)^{k-1}$ blocks

Computed the transversal size of these families

Provided a counterexample to earlier conjectures about maximal families

Abstract

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz established by means of an example that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. L\'{a}szl\'{o} Lov\'{a}sz conjectured that their example is best known example which has the maximum number of blocks. Later it was disproved. But the quest for such examples remain valid till this date. In this short note, by computing transversal size of a certain series of uniform intersecting families constructed over the cycle graph, we provide an example which has more than $(\frac{k}{2})^{k-1}$ (approximately) blocks.