Improving the zero-excess lower bound for exact {K3, K4, K5}-coverings of K33

Determine how far beyond 57 one can improve the lower bound on (33, {3,4,5}, 2), the minimum number of cliques of orders 3, 4, and 5 required to decompose K33 with zero excess (i.e., as an exact decomposition).

Background

For v = 33, general results imply that the minimum excess in a {K3, K4, K5}-cover is zero and provide a general lower bound of 57. The paper focuses on strengthening this lower bound.

Using a layered computational strategy, the paper proves that no 57-block exact decomposition exists and further excludes 58 blocks via counting, establishing (33, {3,4,5}, 2) ≥ 59. The question in the introduction asks how far beyond 57 one can push the zero-excess lower bound, motivating the search that ultimately yields the current best lower bound of 59.