Given a graph H on vertex set {1, 2, · · · , n} and a function f : [0, 1] 2 → R, definewhere µ is the Lebesgue measure on [0, 1]. We say that H is norming if · H is a semi-norm. A similar notion · r(H) is defined by f r(H) := |f | H and H is said to be weakly norming if · r(H) is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate way under the action of a certain natural family of automorphisms is weakly norming. This result includes all previously known examples of weakly norming graphs, but also allows us to identify a much broader class arising from finite reflection groups. We include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture.

A bipartite graph H is said to have Sidorenko's property if the probability that the uniform random mapping from V (H) to the vertex set of any graph G is a homomorphism is at least the product over all edges in H of the probability that the edge is mapped to an edge of G. In this paper, we provide three distinct families of bipartite graphs that have Sidorenko's property. First, using branching random walks, we develop an embedding algorithm which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko's property. Second, we use the concept of locally dense graphs to prove that subdivisions of certain graphs, including cliques, have Sidorenko's property. Third, we prove that if H has Sidorenko's property, then the Cartesian product of H with an even cycle also has Sidorenko's property.

Sidorenko's conjecture states that for every bipartite graph H on {1, · · · , k} (i,j)∈E(H) h(x i , y j )dµ |V (H)| ≥ h(x, y) dµ 2 |E(H)| holds, where µ is the Lebesgue measure on [0, 1] and h is a bounded, nonnegative, symmetric, measurable function on [0, 1] 2 .An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph H to a graph G is asymptotically at least the expected number of homomorphisms from H to the Erdős-Rényi random graph with the same expected edge density as G. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph H with bipartition A ∪ B is tree-arrangeable if neighborhoods of vertices in A have a certain tree-like structure. We show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices a 1 , a 2 in A such that each vertex a ∈ A satisfies N (a) ⊆ N (a 1 ) or N (a) ⊆ N (a 2 ), and also implies a recent result of Conlon, Fox, and Sudakov [3]. Second, if T is a tree and H is a bipartite graph satisfying Sidorenko's conjecture, then it is shown that the Cartesian product T H of T and H also satisfies Sidorenko's conjecture. This result implies that, for all d ≥ 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture.

Given a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing K ′ s,t for the subdivision of the bipartite graph K s,t , we show that ex(n, K ′ s,t ) = O(n 3/2− 1 2s ). This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for t sufficiently large in terms of s. Second, for any integers s, k ≥ 1, we show that ex(n, L) = Θ(n 1+ s sk+1 ) for a particular graph L depending on s and k, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number r ∈ (1, 2) is realisable in the sense that ex(n, H) = Θ(n r ) for some appropriate graph H, giving infinitely many new realisable exponents and implying that 1 + 1/k is a limit point of realisable exponents for all k ≥ 1. Writing H k for the k-subdivision of a graph H, this result also implies that for any bipartite graph H and any k, there exists δ > 0 such that ex(n, H k−1 ) = O(n 1+1/k−δ ), partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph H with maximum degree r on one side which does not contain C 4 as a subgraph satisfies ex(n, H) = o(n 2−1/r ). . s = r. On the other hand, a recent conjecture of Conlon and Lee [5] says that containing K r,r as a subgraph should be the only reason why Theorem 1.1 is tight up to the constant.Conjecture 1.2 (Conlon-Lee). Let H be a bipartite graph such that in one of the parts all the degrees are at most r and H does not contain K r,r as a subgraph. Then there exists some δ > 0 such that ex(n, H) = O(n 2−1/r−δ ).To say more, recall that the k-subdivision of a graph L is the graph obtained by replacing the edges of L by internally disjoint paths of length k + 1. We shall write L k for the ksubdivision of L and L ′ for the 1-subdivision. It is easy to see that any C 4 -free bipartite graph in which every vertex in one part has degree at most two is a subgraph of K ′ t for some positive integer t. Conlon and Lee [5] verified their conjecture in the r = 2 case by proving the following result.Theorem 1.3 (Conlon-Lee). For any integer t ≥ 3, ex(n, K ′ t ) = O(n 3/2−1/6 t ).Our first result gives some small progress towards Conjecture 1.2 when r > 2.Theorem 1.4. Let H be a bipartite graph such that in one of the parts all the degrees are at most r and H does not contain C 4 as a subgraph. Then ex(n, H) = o(n 2−1/r ).

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