Every Monotone 3-Graph Property is Testable

Let k ≥ 2 be a fixed integer and P be a property of k-uniform hypergraphs. In other words, P is a (typically infinite) family of k-uniform hypergraphs and we say a given hypergraph H satisfies P if H ∈ P . For a given constant η > 0 a k-uniform hypergraph H on n vertices is η-far from P if no hypergraph obtained from H by changing (adding or deleting) at most ηn edges in H satisfies P . More precisely, H is η-far from P if no hypergraph G with |E(G)4E(H)| ≤ ηn satisfies P . This is a natural measure of how far the given hypergraph H is to satisfy the property P .