Pure pairs. VIII. Excluding a sparse graph Alex Scott Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Paul Seymour1 Princeton University, Princeton, NJ 08544, USA Sophie Spirkl2 University of Waterloo, Waterloo, Ontario N2L3G1, Canada July 21, 2020; revised June 18, 2021 1Supported by AFOSR grant A9550-19-1-0187 and NSF grant DMS-1800053. 2We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [funding reference number RGPIN-2020-03912]. Cette recherche a ´et´efinanc´eepar le Conseil de recherches en sciences naturelles et en gnie du Canada (CRSNG) [num´erode r´ef´erenceRGPIN-2020-03912]. Abstract A \pure pair" in a graph G is a pair A; B of disjoint subsets of V (G) such that A is complete or anticomplete to B. Let H be a graph, with complement H: is it true that for every graph G (with jGj > 1), if G contains neither H nor H as an induced subgraph, then G has a pure pair A; B in G with jAj; jBj ≥ Ω(jGj)? With Maria Chudnovsky, we answered this in [1]: the answer is \yes" if and only if either H or H is a forest. That leads to the question, let H be a graph, and let 0 < c < 1: is it true that for all G (with jGj > 1), if G contains neither H nor H as an induced subgraph, there is a pure pair A; B in G with jAj; jBj ≥ Ω(jGj1−c)? The answer is related to the congestion of H, the maximum of 1 − (jJj − 1)=jE(J)j over all subgraphs J of H with an edge. (Congestion is nonnegative, and equals zero exactly when H is a forest.) Let d be the smaller of the congestions of H and H. We show that the answer to the question above is \yes" if d ≤ c=(9 + 15c), and \no" if d > c. 1 Introduction Graphs in this paper are finite, and without loops or parallel edges. Let A; B ⊆ V (G) be disjoint. We say that A is complete to B, or A; B are complete, if every vertex in A is adjacent to every vertex in B, and similarly A; B are anticomplete if no vertex in A has a neighbour in B. We say A covers B if every vertex in B has a neighbour in A.A pure pair in G is a pair A; B of disjoint subsets of V (G) such that A; B are complete or anticomplete. The number of vertices of G is denoted by jGj. The complement graph of G is denoted by G. Let us say G contains H if some induced subgraph of G is isomorphic to H, and G is H-free otherwise. If X ⊆ V (G), G[X] denotes the subgraph induced on X. With Maria Chudnovsky, we proved the following in [1]: 1.1 For every forest H there exists " > 0 such that for every graph G with jGj > 1 that is both H-free and H-free, there is a pure pair A; B in G with jAj; jBj ≥ "jGj. It is easy to see (with a random construction) that this has a converse: if H is a graph and neither H nor H is a forest then there is no " as in 1.1. A theorem of Erd}os, Hajnal and Pach [2] (not normally stated in this form, but this is equivalent) says: 1.2 For every graph H there exist "; c > 0 such that for every graph G with jGj > 1 that is both H-free and H-free, there is a pure pair A; B in G with jAj; jBj ≥ "jGjc. But in this, c might be very small, and that raises the question, what sort of graph H will make 1.2 true with c close to 1? The answer is related to \congestion". Let H be a graph. If E(H) 6= ;, we define the congestion of H to be the maximum of 1 − (jJj − 1)=jE(J)j, taken over all subgraphs J of H with at least one edge; and if E(H) = ;, we define the congestion of H to be zero. Thus the congestion of H is always non-negative, and equals zero if and only if H is a forest. It turns out that graphs H where one of H; H has small congestion satisfy 1.2 with a value of c close to 1, while those where both of H; H have large congestion do not. Let us say these two things more precisely. The first of these statements is the main result of the paper, the following: c 1.3 Let c > 0, and let H be a graph such that one of H; H has congestion at most 9+15c . Then there exists " > 0 such that for every graph G with jGj > 1 that is both H-free and H-free, there is a pure pair A; B in G with jAj; jBj ≥ "jGj1−c. The second statement is the following, which we will prove now: 1.4 Let c > 0, and let H be a graph such that H; H both have congestion more than c. There is no " > 0 such that for every graph G with jGj > 1 that is both H-free and H-free, there is a pure pair A; B in G with jAj; jBj ≥ "jGj1−c. Proof. Let J be a subgraph of H with E(J) 6= ; and jJj−1 < (1−c)jE(J)j, and let J 0 be a subgraph of H with E(J 0) 6= ; and jJ 0j−1 < (1−c)jE(J 0)j. Let max((jJj−1)=jE(J)j; (jJ 0j−1)=jE(J 0)j) = 1−c0; so c < c0 < 1. Choose d with c < d < c0. Let " > 0, let n be a large number, let p = nd−1, and let G be a random graph on n vertices, in which every pair of vertices are adjacent independently 1 with probability p. Then (if n is sufficiently large with " given), an easy calculation (which we omit) shows that G almost surely has no pure pair A; B in G with jAj; jBj ≥ ("=2)n1−c. The expected number of induced subgraphs of G isomorphic to J is at most 0 jJj jE(J)j jJj+(d−1)jE(J)j jJ|−(jJ|−1) 1−d 1−(jJ|−1) c −d n p = n ≤ n 1−c0 = n 1−c0 ≤ n=16 since jE(J)j ≥ (jJj − 1)=(1 − c0). Consequently the probability that there are more than n=4 such subgraphs is at most 1=4. Similarly the probability that there are more than n=4 induced subgraphs isomorphic to J 0 is at most 1=4; and so with positive probability, G contains at most n=4 of copies of J, and most n=4 copies of J 0, and has no pure pair A; B in G with jAj; jBj ≥ ("=2)jGj1−c. But then by deleting at most n=2 vertices, we obtain a graph G0 containing neither J nor J 0, and hence containing neither H nor H, and with no pure pair A; B with jAj; jBj ≥ "jG0j1−c (since "jG0j1−c ≥ ("=2)n1−c). This proves 1.4. A result similar to 1.3 was proved in an earlier paper [4]; its conclusion was somewhat stronger, but it applied only to a more restricted class of graphs. We say that H has branch-length at least k if H is an induced subgraph of a graph that can be constructed as follows: start with a multigraph (possible with loops or parallel edges) and subdivide each edge at least k − 1 times. (It is easy to see that graphs with branch-length at least k have congestion at most 1=k.) We proved in [4] that: 1.5 Let c > 0 with 1=c an integer, and let H be a graph with branch-length at least 4c−1 + 5. Then there exists " > 0 such that for every graph G with jGj > 1 that is both H-free and H-free, there is a pure pair A; B in G with jAj ≥ "jGj and jBj ≥ "jGj1−c. Forests can have small branch-length, so this does not contain 1.1; but we have also proved (unpublished) a similar theorem, that if H can be obtained from a multigraph H0 by selecting a spanning tree and subdividing many times all edges not in the tree, then something like 1.5 holds. (Not quite the analogue of 1.5: for a given value of c, the number of times the non-tree edges have to be subdivided depends not only on c but also on the graph H0.) These graphs also have small congestion, but not every graph with small congestion can be built this way either; for instance, if we take a long cycle, and for each of its vertices v add a new vertex adjacent only to v, this graph has small congestion and is not contained in any graph that can be built in the way described. That raises the question, can 1.5 be extended to all graphs of small congestion, giving a stronger version of 1.3? In other words: 1.6 Possibility: For all c > 0, there exists ξ > 0 with the following property. For every graph H with congestion at most ξ, there exists " > 0 such that for every graph G with jGj > 1 that is H-free and H-free, there is a pure pair A; B in G with jAj ≥ "jGj and jBj ≥ "jGj1−c.