The lower-bound problem for regular induced subgraphs of type-based random graphs

Abstract

For a graph G let F(G) denote the largest order of a regular induced subgraph of G, and let f(n) = min{F(G) : |V(G)| = n}. A problem of Erdos, Fajtlowicz and Staton asks whether f(n)/log n -> infinity. Dyson and McKay have recently proved f(n) <= (sqrt(2e)+o(1)) sqrt(n) via a type-based random model (arXiv:2604.08215). This paper concerns the opposite direction within the type-based family. We conjecture that every model of the family satisfies F(G) >= (sqrt(2e)-o(1)) sqrt(n) asymptotically almost surely, so that sqrt(2e) is the optimal constant obtainable from the family, and we prove the corresponding statement at exponent level -- F(G) >= n^{1/2-eps} -- conditionally on two explicitly stated hypotheses: a local limit lower bound for inhomogeneous degree sequences, and a correlation estimate at sublinear overlaps. The complementary overlap range, including full overlap, requires no correlation hypothesis. We further record the exact-curvature first-moment computation that independently identifies sqrt(2e), including a uniform trace bound on its determinant correction, and certified exact computations at orders up to 48 consistent with the predicted crossover F ~ min(n^{2/3}, sqrt(n/L)). This version substantially revises v1; see the note in Section 1.