Brandt Conjecture

The Brandt conjecture on the binding number asserts that, for all sufficiently large integers , every nonempty simple graph containing no complete graph as a subgraph satisfies

Here is the binding number.

Lin and Yang (2026) proved the bound for every . It is sharp, since the complete multipartite graph with equally sized nonempty parts has binding number . The proposed bound fails for , while the paper does not settle .