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In 1975, P. Erd\"{o}s 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. In this paper, it is proved that $$f(n)\geq n+\frac{107}{3}t+\frac{7}{3}$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq \frac{2119}{4}t^{2}+87978t+\frac{15957}{4}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{7654}{19071}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2.4}$ (Lai, 2003). The conjecture $\lim_{n \rightarrow \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}$ is not true.

In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero

It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions.

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]).