Beame et al. [ITCS 2018 & TALG 2021] introduced and used the Bipartite Independent Set (BIS) and Independent Set (IS) oracle access to an unknown, simple, unweighted and undirected graph and solved the edge estimation problem. The introduction of this oracle set forth a series of works in a short span of time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC 2018], Dell, Lapinskas and Meeks [SODA 2020], Bhattacharya et al. [ISAAC 2019 & Theory Comput. Syst. 2021], and Chen et al. [SODA 2020]. Edge estimation using BIS can be done using polylogarithmic queries, while IS queries need sub-linear but more than polylogarithmic queries. Chen et al. improved Beame et al.'s upper bound result for edge estimation using IS and also showed an almost matching lower bound. Beame et al. in their introductory work asked a few open questions out of which one was on estimating structures of higher order than edges, like triangles and cliques, using BIS queries. In this work, we completely resolve the query complexity of estimating triangles using BIS oracle. While doing so, we prove a lower bound for an even stronger query oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi, Chakrabarty and Khanna [ESA 2021] to test graph connectivity.

翻译：Beame等人[ITCS 2018 & TALG 2021]引入并使用了二分独立集（BIS）与独立集（IS）预言机来访问未知的简单无向无权图，并解决了边估计问题。这一预言机的提出在短时间内引发了一系列研究工作，这些工作要么解决了Beame等人提出的开放问题，要么是其工作的推广，如Dell和Lapinskas [STOC 2018]、Dell、Lapinskas和Meeks [SODA 2020]、Bhattacharya等人[ISAAC 2019 & Theory Comput. Syst. 2021]以及Chen等人[SODA 2020]的研究。利用BIS进行边估计可通过多对数次查询完成，而IS查询则需要次线性但多于多对数的查询次数。Chen等人改进了Beame等人关于使用IS进行边估计的上界结果，并给出了近乎匹配的下界。Beame等人在其开创性工作中提出了若干开放问题，其中之一是如何使用BIS查询估计边之外的高阶结构（如三角形和团）。在本工作中，我们完整解决了使用BIS预言机估计三角形的查询复杂度问题。在此过程中，我们针对一种更强的查询预言机——边空查询（EE）预言机——证明了下界，该预言机由Assadi、Chakrabarty和Khanna [ESA 2021]近期引入，用于测试图的连通性。