Fully Computer-Assisted Proofs in Extremal Combinatorics

We present a fully computer-assisted proof system for solving particular family of problems in Extremal Combinatorics. Existing techniques using Flag Algebras have proven powerful the past, but so far lacked computational counterpart to derive matching constructive bounds. demonstrate that common search heuristics are capable finding constructions beyond reach human intuition. Additionally, most obvious downside such heuristics, namely missing guarantee global optimality, can often be eliminated this case through lower bounds and stability results coming from Algebra approach. To illustrate potential approach, we study two related well-known Graph Theory go back questions Erdős 60s. Most notably, first major improvement upper bound Ramsey multiplicity K_4 25 years, precisely determine off-diagonal number, settle minimum number independent sets size four graphs with clique strictly less than five.