Minimum degree and the graph removal lemma

Abstract

The clique removal lemma says that for every r ≥ 3 $r\ge 3$ and ε > 0 $\varepsilon \gt 0$, there exists some δ > 0 $\delta \gt 0$ so that every n $n$‐vertex graph G $G$ with fewer than δ n r $\delta {n}^{r}$ copies of K r ${K}_{r}$ can be made K r ${K}_{r}$‐free by removing at most ε n 2 $\varepsilon {n}^{2}$ edges. The dependence of δ $\delta $ on ε $\varepsilon $ in this result is notoriously difficult to determine: it is known that δ − 1 ${\delta }^{-1}$ must be at least super‐polynomial in ε − 1 ${\varepsilon }^{-1}$, and that it is at most of tower type in log ε − 1 $\mathrm{log} {\varepsilon }^{-1}$. We prove that if one imposes an appropriate minimum degree condition on G $G$, then one can actually take δ $\delta $ to be a linear function of ε $\varepsilon $ in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super‐polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.

The Minimum Degree Removal Lemma Thresholds

“…Theorem 1.2. δ poly-rem (C 2k+1 ) = 1 2k+1 . Theorem 1.2 also answers another question of Fox and Wigderson [12], of whether δ lin-rem (H) and δ poly-rem (H) can only obtain finitely many values on r-chromatic graphs H for a given r ≥ 3. Theorem 1.2 shows that δ poly-rem (H) obtains infinitely many values for 3-chromatic graphs.…”

“…Here we prove the lower bounds in Theorems 1.2 and 1.3. The lower bound in Theorem 1.2 was proved in [12,Theorem 4.3]. For completeness, we include a sketch of the proof: Lemma 3.1. δ poly-rem (C 2k+1 ) ≥ 1 2k+1 .…”

“…However, it is known [2] that δ(ε, H) is only polynomial in ε if H is bipartite. This situation led Fox and Wigderson [12] to initiate the study of minimum degree conditions which guarantee that δ(ε, H) depends polynomially or linearly on ε. Formally, let δ(ε, H; γ) be the maximum δ ∈ [0, 1] such that if G is an n-vertex graph with minimum degree at least γn and with εn 2 edge-disjoint copies of H, then G contains δn v(H) copies of H. Definition 1.1.…”

“…Trivially, δ lin-rem (H) ≥ δ poly-rem (H). Fox and Wigderson [12] initiated the study of δ lin-rem (H) and δ poly-rem (H), and proved that δ lin-rem (K r ) = δ poly-rem (K r ) = 2r−5 2r−3 for every r ≥ 3, where K r is the clique on r vertices. They further asked to determine the removal lemma thresholds of odd cycles.…”

“…, C 2k+1 }) = 1 2k+1 . The lower bound in Theorem 1.2 was established in [12]; for completeness, we sketch the proof in Section 3.…”