A function $${f : {\mathbb F}_{2}^{n} \rightarrow {\{0,1\}}}$$f:F2n→{0,1} is triangle-free if there are no $${x_{1},x_{2},x_{3} \in {\mathbb F}_{2}^{n}}$$x1,x2,x3∈F2n satisfying $${x_{1} + x_{2} + x_{3} = 0}$$x1+x2+x3=0 and $${f(x_{1}) = f(x_{2}) = f(x_{3}) = 1}$$f(x1)=f(x2)=f(x3)=1. In testing triangle-freeness, the goal is to distinguish with high probability triangle-free functions from those that are $${\varepsilon}$$ε-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 $${\varepsilon}$$ε (Green 2005); however, the best-known upper bound is a tower-type function of $${1/\varepsilon}$$1/ε. The best known lower bound on the query complexity of the canonical tester is $${1/\varepsilon^{13.239}}$$1/ε13.239 (Fu & Kleinberg 2014).In this work we introduce a new approach to proving lower bounds on the query complexity of triangle-freeness. We relate the problem to combinatorial questions on collections of vectors in $${{\mathbb Z}_D^n}$$ZDn and to sunflower conjectures studied by Alon, Shpilka & Umans (2013). The relations yield that a refutation of the Weak Sunflower Conjecture over $${{\mathbb Z}_{4}}$$Z4 implies a super-polynomial lower bound on the query complexity of the canonical tester for triangle-freeness. Our results are extended to testing k-cycle-freeness of functions with domain $${{\mathbb F}_p^n}$$Fpn for every $${k \ge 3}$$k≥3 and a prime p. In addition, we generalize the lower bound of Fu and Kleinberg to k-cycle-freeness for $${k \geq 4}$$k≥4 by generalizing the construction of uniquely solvable puzzles due to Coppersmith & Winograd (1990).

中文翻译：

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向日葵和测试三角形-函数的自由度

一个函数 $${f : {\mathbb F}_{2}^{n} \rightarrow {\{0,1\}}}$$f:F2n→{0,1} 如果有否 $${x_{1},x_{2},x_{3} \in {\mathbb F}_{2}^{n}}$$x1,x2,x3∈F2n 满足 $${x_{1 } + x_{2} + x_{3} = 0}$$x1+x2+x3=0 和 $${f(x_{1}) = f(x_{2}) = f(x_{3}) = 1}$$f(x1)=f(x2)=f(x3)=1。在测试三角形自由度时，目标是将高概率的无三角形函数与那些 $${\varepsilon}$$ε-远非无三角形函数区分开来。Green 表明，该问题的规范测试器的查询复杂性的上限是一个仅依赖于 $${\varepsilon}$$ε 的函数（Green 2005）；然而，最著名的上限是 $${1/\varepsilon}$$1/ε 的塔型函数。规范测试器查询复杂性的最著名下限是 $${1/\varepsilon^{13.239}}$$1/ε13.239 (Fu & Kleinberg 2014)。在这项工作中，我们引入了一种新方法来证明三角形自由度的查询复杂度的下限。我们将这个问题与 $${{\mathbb Z}_D^n}$$ZDn 中向量集合的组合问题以及 Alon、Shpilka & Umans (2013) 研究的向日葵猜想联系起来。该关系产生了对 $${{\mathbb Z}_{4}}$$Z4 的弱向日葵猜想的反驳，这意味着三角自由度的规范测试器的查询复杂度的超多项式下界。我们的结果扩展到测试具有域 $${{\mathbb F}_p^n}$$Fpn 的函数的 k 循环自由度，对于每个 $${k \ge 3}$$k≥3 和一个素数 p。此外，我们通过对由于 Coppersmith & Winograd (1990) 提出的唯一可解谜题的构造进行推广，将 Fu 和 Kleinberg 的下界推广到 k-cycle-freeness for $${k \geq 4}$$k≥4。