Tianchi Yang

How many graphs on n vertices contain no four-cycle was known only up to a constant factor in the exponent, with a gap between the Kleitman–Winston upper bound and the matching lower bound. The count is now proved to be $2^{O(n^{3/2})}$, answering a question of Erdős. The proof applies the hypergraph container method only to four-cycles whose two diagonals both have small codegree, defers the rest to counting inside each container, and adds a new bound on the number of matchings in a graph.

How many graphs on n vertices contain no four-cycle was known only up to a constant factor in the exponent, with a gap between the Kleitman–Winston upper bound and the matching lower bound. The count is now proved to be $2^{O(n^{3/2})}$, answering a question of Erdős. The proof applies the hypergraph container method only to four-cycles whose two diagonals both have small codegree, defers the rest to counting inside each container, and adds a new bound on the number of matchings in a graph.