The Kohayakawa-Kreuter conjecture concerns when random graphs exhibit Ramsey properties for families of graphs. The conjecture identifies a threshold function — a precise density at which the random graph transitions from almost surely lacking the property to almost surely having it. Below the threshold, you can color the edges to avoid every pattern in the family. Above it, every coloring contains at least one.
The obstruction to proving the conjecture has been structural: showing that sparse graphs can be decomposed in specific ways. Kuperwasser, Samotij, and Wigderson conjectured that any (m,0)-sparse graph — one where every subgraph has at most m edges per vertex — can be split into a (1,1)-sparse part and an (m,2m-1)-sparse part. This decomposition lemma, if true, would supply the key structural ingredient for the broader Ramsey threshold result.
Yancey proves it. The decomposition holds universally: every graph satisfying the global sparsity condition can be partitioned into two pieces with the required local properties. The global constraint propagates downward into a pair of tighter local constraints, and the partition achieving this always exists.
The through-claim is about what sufficiency means in combinatorics. The Kohayakawa-Kreuter conjecture is a statement about random graphs and probabilistic thresholds. The tool that resolves it is a deterministic decomposition theorem about sparse graphs. The randomness was never the hard part. The hard part was proving that a structural property — the existence of a specific partition — holds for all graphs in the class, not just for typical ones. The probabilistic statement rested on a deterministic foundation that took decades to establish.