2026/07/21 by Shuang Sun, Yan Wang, Yuyao Yang +1
#math.CO
Fix an integer r≥ 2 and a finite simple r-uniform hypergraph F with at least one edge and no isolated vertices. An n-vertex r-graph is ε-far from being F-free if at least εnr edges must be deleted to destroy every copy of F. A finite r-graph H is F-abundant if there are constants c,C>0 such that every sufficiently large ε-far host contains at least cεC nv(H) labelled copies of H. A family is F-abundant when one member has this lower bound in each host, although the member may depend on the host and on ε, while c and C are common to the family. We prove that every abundant family contains an abundant member. We prove the analogous coloured theorem for F-partite hosts containing edge-disjoint part-respecting copies of F such that every vertex lies in at least εnr-1 of them. The case r=2 yields the coloured and uncoloured graph compactness theorems, answers Question 5.2 of Girão, Hurley, Illingworth and Michel, and proves their Conjecture 5.1 [J. Lond. Math. Soc., 2024]. We also obtain an explicit bound \lfloor 2r(C+1)\rfloor for the order of the non-isolated core of a selected witness. Moreover, we give several applications. For example, we construct translation-invariant linear systems from abundant coloured hypergraphs, obtain a square-root bound for an equation associated with a cycle of bounded length, give a one-sided tester based on one fixed graph when distance from the property gives a polynomial lower bound on distance from being F-free, and prove that no algorithm decides whether a family of finite simple graphs enumerated by a Turing machine is K3-abundant.