It is shown that if a function$u$satisfies a backward parabolic inequality in an open set Ω∉$R$^{$n$+1}and vanishes to infinite order at a point ($x$_{0}·$t$_{0}) in Ω, then$u(x, t$_{0})=0 for all$x$in the connected component of$x$_{0}in Ω⌢($R$^{$n$}×{$t$_{0}}).

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]).

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