On the Duke--Erdős--Rödl Problem at the One-Third Threshold

Abstract: Let $G$ be an $n$-vertex graph with $e(G)\ge n<sup>2/k$. We prove a self-contained internal short-cycle core theorem at the threshold $k\le n<sup>{1/3}$: the graph $G$ contains a subgraph $H_6$ with $Ω(n<sup>2/k<sup>3)$ edges in which every two distinct edges lie together on a cycle of length at most $6$ contained in $H_6$, and a subgraph $H_8$ with $Ω(n<sup>2/k<sup>2)$ edges in which every two distinct edges lie together on a cycle of length at most $8$ contained in $H_8$. In density notation $ρ=e(G)/n<sup>2$, this gives internal cores of sizes $Ω(ρ<sup>3n<sup>2)$ and $Ω(ρ<sup>2n<sup>2)$ throughout the range $ρ\ge n<sup>{-1/3}$. The $C_{\le6}$ conclusion above is an edge-connected statement and does not impose the adjacent-edge $C_4$ condition appearing in the strongest Duke--Erdős--Rödl formulation. We also include two complementary results clarifying this distinction. First, under the ambient-witness convention, every graph with at least $n<sup>2/k$ edges and $k=o(n<sup>{1/2})$ contains $Ω(n<sup>2/k<sup>3)$ selected edges whose pairs are witnessed by ambient cycles of length at most $6$, with adjacent pairs witnessed by ambient $C_4$'s. Second, under the standard internal strong $C_6$ convention, for every fixed $β\in[1/3,1/2)$ there is an infinite sequence of bipartite graphs $G$ with $n\to\infty$ and $e(G)=Θ<em>β(n<sup>{2-β})$ such that every internally strongly $C_6$-connected subgraph has only $$O</em>β(ρ(G)^{3n^2/(\log} n)<sup>2)$$ edges. The obstruction is a random cyclic shift-lift of $K_{q,q}$, together with an occupancy estimate excluding large aligned two-covers.