Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, March 23, 2023, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Zhihan Jin

The Minimum Degree Removal Lemma Thresholds

The graph removal lemma is a fundamental result in extremal graph theory, which says that for every fixed graph H and &#x3B5>0, if an n-vertex graph G contains &#x3B5n^2 edge-disjoint copies of H then G contains &#x3B4n^{v(H)} copies of H for some &#x3B4=&#x3B4(&#x3B5,H)>0. The current proofs of the removal lemma give only very weak bounds on &#x3B4(&#x3B5,H), and it is also known that &#x3B4(&#x3B5,H) is not polynomial in &#x3B5 unless H is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that &#x3B4(&#x3B5,H) depends polynomially or linearly on &#x3B5. In this paper we answer several questions of Fox and Wigderson on this topic. In particular, we provide the polynomial removal lemma threshold for odd cycles and give the full characterization of the linear removal lemma threshold for 3-chromatic graphs.

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