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Let $F$ be a bipartite graph and $\delta_F=\delta(F)$ be a constant. Let $p\geq 0$ be a given constant. Let $W$ be a graphon with $\int W=p$ and for any $A,B\subseteq \left[0,1\right]$ with $|A|,|B|\g …

Let $0\leq d\ll\varepsilon,\frac{1}{e},\frac{1}{v}\leq 1.$ Let $G$ be a $n$-vertices graph ($n$ is sufficient large, $1/n\ll d$) and for any $A,B\subseteq V(G)$, the edge density $d(A,B)\geq d.$ Then …

Does the following holds? For every bipartite graph $H$ and every graph $G$ with $\frac{\Delta(G)}{\delta(G)}\leq 2$, $$t(H,G)\geq t(K_2, G)^{e(H)}.$$ If not sure, is this a equal question as Sidorenk …

Recently I am reading a paper (https://arxiv.org/abs/1504.00858) with respect to edge-vertex transitive graphs. What is the property of the graph that is edge transitive and vertex transitive? I know …

Given a positive constant $0<\gamma<1$, does there exists integer $k_0>0$ such that for any integer $k\geq k_0$ the following holds?: For any $A\subseteq\left[0,1\right]^k$ with the measure of $A$ sat …

For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph. For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it …

Does the following hold? For every bipartite graph $H$ and every graph $G$ with $e(G)\geq 0.1(v(G))^2$, $$t(H,G)\geq t(K_2, G)^{e(H)}.$$ If not sure, is this a equal question as Sidorenko's conjecture …

A graphon is a measurable symmetric function $W: [0,1]\to [0,1].$ By Lovasz's book "Large networks and graph limits" we know for any graph sequence $G_1, G_2, \dots G_i,\dots$ there exists a graphon $ …

Let $G$ be a bipartite graph (vertex number sufficient large) with bipartition $(U,W)$ and edge density $d$. Does these two statement equal? $G$ is $\varepsilon$-regular, i.e. $\big|e_G(X,Y)-d|X||Y|\ …

Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions $W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert …

Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions $W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert …

Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where …

Given two graphs $H$ and $G$, the homomorphism density $t(H, G)$ is defined as the proportion of mappings from the vertices of $H$ to the vertices of $G$ that preserve adjacency. Formally, $$ t(H, …

I would like to ask a question about Sidorenko's conjecture. Here is the background of my question: Quasi-random graphs A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain prop …