In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq 540t^{2}+\frac{175811}{2}t+\frac{7989}{2}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2 \over 5}},$ which is better than the previous bounds $\sqrt 2$ (see [2]), $\sqrt {2+{2562\over 6911}}$ (see [7]).

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by$δ$are called$δ$-$uniform$. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.

A simplicial complex Δ is called$flag$if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension$d$−1, then the graph of Δ (i) is (2$d$−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the$d$-dimensional cross-polytope. Second, the$h$-vector of a flag simplicial homology sphere Δ of dimension$d$−1 is minimized when Δ is the boundary complex of the$d$-dimensional cross-polytope.

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If$P$is a simplicial$d$-polytope then its$h$-vector ($h$_{0},$h$_{1}, …,$h$_{$d$}) satisfies $$ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $$ . Moreover, if$h$_{$r$−1}=$h$_{$r$}for some $$ r\leq \frac{1}{2}d $$ then$P$can be triangulated without introducing simplices of dimension ≤$d$−$r$.

An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.