Regular subgraphs at every density

TL;DR

The paper settles, up to absolute constants, the Erdős–Sauer problem by proving that graphs on $n$ vertices without an $r$-regular subgraph have at most $C r^2 n \log\log n$ edges for fixed $r$, and extends the threshold to the full range of $r$ with a dichotomy: every $n$-vertex graph with average degree at least $\min\big(C r\log(n/r), C r^2 \log\log n\big)$ contains an $r$-regular subgraph. A central tool is a novel near-regularity process that extracts large subgraphs whose degree variance is tightly controlled, enabling a reduction to (near-)regular cases where the Alon–Friedland–Kalai regularity results apply. The authors also develop a sunflower-based hypergraph method to convert certain regularity-type configurations into actual $r$-regular subgraphs, and provide tight lower-bound constructions showing the phase transition around $r \approx \log n$. Beyond the Erdős–Sauer regime, the paper proves a general regularity-in-almost-regular graphs theorem, and demonstrates tightness via probabilistic constructions for both small and large $r$, thereby resolving the problem up to a universal constant factor and outlining avenues for hypergraph extensions.

Abstract

In 1975, Erdős and Sauer asked to estimate, for any constant $r$, the maximum number of edges an $n$-vertex graph can have without containing an $r$-regular subgraph. In a recent breakthrough, Janzer and Sudakov proved that any $n$-vertex graph with no $r$-regular subgraph has at most $C_r n \log \log n$ edges, matching an earlier lower bound by Pyber, Rödl and Szemerédi and thereby resolving the Erdős-Sauer problem up to a constant depending on $r$. We prove that every $n$-vertex graph without an $r$-regular subgraph has at most $Cr^2 n \log \log n$ edges. This bound is tight up to the value of $C$ for $n\geq n_0(r)$ and hence resolves the Erdős-Sauer problem up to an absolute constant. Moreover, we obtain similarly tight results for the whole range of possible values of $r$ (i.e., not just when $r$ is a constant), apart from a small error term at a transition point near $r\approx \log n$, where, perhaps surprisingly, the answer changes. More specifically, we show that every $n$-vertex graph with average degree at least $\min(Cr\log(n/r),Cr^2 \log\log n)$ contains an $r$-regular subgraph. The bound $Cr\log(n/r)$ is tight for $r\geq \log n$, while the bound $Cr^2 \log \log n$ is tight for $r<(\log n)^{1-Ω(1)}$. These results resolve a problem of Rödl and Wysocka from 1997 for almost all values of $r$. Among other tools, we develop a novel random process that efficiently finds a very nearly regular subgraph in any almost-regular graph. A key step in our proof uses this novel random process to show that every $K$-almost-regular graph with average degree $d$ contains an $r$-regular subgraph for some $r=Ω_K(d)$, which is of independent interest.

Theorems & Definitions (31)

• Theorem 1.1: Pyber pyber1985regular
• Theorem 1.2: Pyber--Rödl--Szemerédi PRSz95
• Theorem 1.3: Janzer--Sudakov janzer2023resolution
• Theorem 1.4
• Theorem 1.5
• Proposition 1.6
• Proposition 1.6
• Definition 1.7
• Lemma 1.8: Alon et al. ARS+17 and Bucić et al. bucic2020nearly
• Proposition 1.9