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Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots ,k\}$ \begin{eqnarray*} \int \prod _{(i,j)\in E(H)} h(x_i, y_j) d\mu ^{|V(H)|} \ge \left ( \int h(x,y) d\mu ^2 \right )^{|E(H)|} \end{eqnarray*} holds, where $\mu$ is the Lebesgue measure on $[0,1]$ and $h$ is a bounded, non-negative, 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 asymptoticall

A celebrated conjecture of Sidorenko and Erdős-Simonovits states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as t

Abstract A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. Burr and Rosta, extending a famous conjecture of Erdős, conjectured that every graph is common. The conjectures of Erdős and of Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples of common graphs had not seen much progress since then, although very recent

Sidorenko's conjecture for blow-ups, Discrete Analysis 2021:2, 13 pp. Let $G$ be a bipartite graph with finite vertex sets $X$ and $Y$. If $G$ has density $\alpha$, then the average degree of the vertices in $X$ is $\alpha|Y|$, so the mean-square degree is at least $\alpha^2|Y|^2$. This is easily seen to be equivalent to the statement that if two vertices $y_1,y_2$ are selected independently and uniformly at random from $Y$, then the average number of neighbours they have in common in $X$ is at

Abstract Majority dynamics on a graph is a deterministic process such that every vertex updates its ‐assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnell, Tamuz and Tan conjectured that, in the Erdős–Rényi random graph , the random initial ‐assignment converges to a ‐agreement with high probability whenever . This conjecture was first confirmed for for a large constant by Fountoulakis, Kang and Makai. Although this result has bee

Abstract We prove that every properly edge-colored n -vertex graph with average degree at least $$32(\log 5n)^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>32</mml:mn> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mn>5</mml:mn> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> contains a rainbow cycle, improving upon the $$(\log n)^{2+o(1)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/

Abstract One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich, and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a constant $C$ such that every graph with $n$ vertices and $C n^{2 - 1/r}$ edges contains a copy of $H$. This result is tight up to the constant when $H$ contains a copy of $K_{r,s}$ with $s$ sufficiently large in terms of $r$. We conjecture that this i

This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader class, comparable to that given in work of Szegedy, through further iteration.

For any given graph H $H$ , one may define a natural corresponding functional ∥ . ∥ H $\Vert .\Vert _H$ for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once H $H$ is paired with a 2-edge-colouring α $\alpha$ to assign conjugates. We say that H $H$ is real-norming (respectively complex-norming) if ∥ . ∥ H $\Vert .\Vert _H$ (respectively ∥ . ∥ H , α $\Vert .\Vert _{H,\alpha }$ for some α $\alpha$ ) is a norm on the vector space of real

Abstract Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given $\mu&gt;0$ , there exists $n_0$ such that the following holds. Every n -vertex Steiner triple system contains all hypertrees with at most $(1-\mu )n$ vertices whenever $n\geq n_0$ . We prove this conjecture.