A function f:F2n→{0,1}\documentclass[12pt]{minimal}
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\begin{document}$${f : {\mathbb F}_{2}^{n} \rightarrow {\{0,1\}}}$$\end{document} is triangle-free if there are no x1,x2,x3∈F2n\documentclass[12pt]{minimal}
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\begin{document}$${x_{1},x_{2},x_{3} \in {\mathbb F}_{2}^{n}}$$\end{document} satisfying x1+x2+x3=0\documentclass[12pt]{minimal}
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\begin{document}$${x_{1} + x_{2} + x_{3} = 0}$$\end{document} and f(x1)=f(x2)=f(x3)=1\documentclass[12pt]{minimal}
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\begin{document}$${f(x_{1}) = f(x_{2}) = f(x_{3}) = 1}$$\end{document}. In testing triangle-freeness, the goal is to distinguish with high probability triangle-free functions from those that are ε\documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon}$$\end{document}-far from being triangle-free. It was shown by Green that the query complexity of the canonical tester for the problem is upper bounded by a function that depends only on ε\documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon}$$\end{document} (Green 2005); however, the best-known upper bound is a tower-type function of 1/ε\documentclass[12pt]{minimal}
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\begin{document}$${1/\varepsilon}$$\end{document}. The best known lower bound on the query complexity of the canonical tester is 1/ε13.239\documentclass[12pt]{minimal}
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\begin{document}$${1/\varepsilon^{13.239}}$$\end{document} (Fu & Kleinberg 2014).
