Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the size of graphs without repeated cycle lengths   In 1975, P. Erd\"os 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 $$f(n)\geq
n+\frac{107}{3}t+\frac{7}{3}$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq
\frac{2119}{4}t^{2}+87978t+\frac{15957}{4}$. Consequently, $\lim \inf_{n \to
\infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{7654}{19071}},$ which is
better than the previous bounds $\sqrt 2$ [Y. Shi, Discrete Math. 71(1988),
57-71], $\sqrt {2.4}$ [C. Lai, Australas. J. Combin. 27(2003), 101-105]. The
conjecture $\lim_{n \rightarrow \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}$ is
not true.

6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga