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{\bf Thomas Lam and Jacques Verstra\"{e}te}
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{\bf A Note on Graphs Without Short Even Cycles}
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In this note, we show that any $n$-vertex graph
without even cycles of length at most $2k$ has at most
${1\over2}n^{1 + 1/k} + O(n)$ edges, and
polarity graphs of generalized polygons show that
this is asymptotically tight when $k \in \{2,3,5\}$.

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