Non-canonical maximum cliques without a design structure in the block graphs of 2-designs

The article delves into the study of maximum cliques in the block graphs of 2-designs, specifically focusing on non-canonical cliques that do not adhere to a design structure. It addresses a problem posed by Chris Godsil and Karen Meagher, providing a counterexample to their conjecture about the isomorphism of non-canonical cliques to smaller designs. The authors present a 2-(66,6,1) design with exactly 14 non-canonical maximum cliques, none of which have a design structure. This discovery challenges the previously held belief and opens up new avenues for research in combinatorial design theory and graph theory.