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- W2951995695 abstract "Ajtai, Koml'os, and Szemer'edi proved that for sufficiently large $t$ every triangle-free graph with $n$ vertices and average degree $t$ has an independent set of size at least $frac{n}{100t}log{t}$. We extend this by proving that the number of independent sets in such a graph is at least [ 2^{(1/2400)frac{n}{t}log^2{t}}. ] This result is sharp for infinitely many $t,n$ apart from the constant. An easy consequence of our result is that there exists $c'>0$ such that every $n$-vertex triangle-free graph has at least [ 2^{c'sqrt n log n} ] independent sets. We conjecture that the exponent above can be improved to $sqrt{n}(log{n})^{3/2}$. This would be sharp by the celebrated result of Kim which shows that the Ramsey number $R(3,k)$ has order of magnitude $k^2/log k$." @default.
- W2951995695 created "2019-06-27" @default.
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- W2951995695 date "2011-11-16" @default.
- W2951995695 modified "2023-09-27" @default.
- W2951995695 title "Counting independent sets in triangle-free graphs" @default.
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