We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on Λ(x,t)=(|∇T(x,t)|^2g(t)+|Rm(x,t)|^2g(t))^{1/2} will imply bounds on all covariant derivatives of Rm and T. (2). We show that Λ(x,t) will blow up at a finite-time singularity, so the flow will exist as long as Λ(x,t) remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups$G$which possess certain “density” properties (see$Moskowitz$, M., On the density theorems of Borel and Furstenberg,$Ark. Mat.$$16$(1978), 11–27, and$Moskowitz$, M., Some results on automorphisms of bounded displacement and bounded cocycles,$Monatsh. Math.$$85$(1978), 323–336). For such groups we show that Γ always has finite index in its normalizer$N$_{$G$}(Γ). We then investigate analogous questions for the automorphism group Aut($G$) proving, under appropriate conditions, that Stab_{Aut($G$)}(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$ , of Aut($G$) has finite index in Stab_{Aut($G$)}(Γ). We test the limits of our results with various examples and counterexamples.

In 1975, P. Erd\H os proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. The sequence $(c_1,c_2,\cdots,c_n)$ is the cycle length distribution of a graph $G$ of order $n$， where $c_i$ is the number of cycles of length $i$ in $G$. Let $f(a_1,a_2,\cdots,$ $a_n)$ denote the maximum possible number of edges in a graph which satisfies $c_i\leq a_i$ where $a_i$ is a nonnegative integer. In 1991, Shi posed the problem of determining $f(a_1,a_2,\cdots,a_n)，$ which extended the problem due to Erd\H os, it is clear that $f(n)=f(1,1,\cdots,1)$. Let $g(n,m)=f(a_1,a_2,\cdots,a_n),$ $a_i=1$ for all $i/m$ be integer, $a_i=0$ for all $i/m$ be not integer. It is clear that $f(n)=g(n,1)$. We prove that $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{40}{99}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2 + \frac{7654}{19071}}$ (Lai, 2017). We show that $\liminf_{n \rightarrow \infty} {g(n,m)-n\over \sqrt \frac{n}{m}} > \sqrt {2.444},$ for all even integers $m$. We make the following conjecture: $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} > \sqrt {2.444}.$

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