## Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs

##### Abstract

A graph is H-free if it has no induced subgraph isomorphic to H, and |G| denotes the number
of vertices of G. A conjecture of Conlon, Sudakov and the second author asserts that:
- For every graph H, there exists ∈ > 0 such that in every H-free graph G with |G| > 1 there
are two disjoint sets of vertices, of sizes at least ∈|G|∈ and ∈|G|, complete or anticomplete to
each other.
This is equivalent to:
- The "sparse linear conjecture": For every graph H, there exists ∈ > 0 such that in every
H-free graph G with |G| > 1, either some vertex has degree at least ∈|G|, or there are two
disjoint sets of vertices, of sizes at least ∈|G|∈ and ∈|G|, anticomplete to each other.
We prove a number of partial results towards the sparse linear conjecture. In particular, we prove
it holds for a large class of graphs H, and we prove that something like it holds for all graphs H.
More exactly, say H is "almost-bipartite" if H is triangle-free and V (H) can be partitioned into
a stable set and a set inducing a graph of maximum degree at most one. (This includes all graphs
that arise from another graph by subdividing every edge at least once.) Our main result is:
- The sparse linear conjecture holds for all almost-bipartite graphs H.
(It remains open when H is the triangle K3.) There is also a stronger theorem:
- For every almost-bipartite graph H, there exist ∈; t > 0 such that for every graph G with
|G| > 1 and maximum degree less than ∈|G|, and for every c with 0 < c ≤ 1, either G
contains ∈ct|G||H| induced copies of H, or there are two disjoint sets A,B V (G) with
|A| ≥ ∈ct|G| and |B| ≥ ∈|G| and with at most c|A| . |B| edges between them.
We also prove some variations on the sparse linear conjecture, such as:
- For every graph H, there exists ∈ > 0 such that in every H-free graph G with |G| > 1
vertices, either some vertex has degree at least ∈|G|, or there are two disjoint sets A, B of
vertices with |A| . |B| ≥ ∈|G|1+∈, anticomplete to each other.

##### Collections

##### Cite this version of the work

Maria Chudnovsky, Jacob Fox, Alex Scott, Paul Seymour, Sophie Spirkl
(2020).
Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs. UWSpace.
http://hdl.handle.net/10012/18509

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