made some macro changes

atri.bib

@@ -0,0 +1,21 @@
+@inproceedings{triang-hard,
+  author    = {Tsvi Kopelowitz and
+               Virginia Vassilevska Williams},
+  editor    = {Artur Czumaj and
+               Anuj Dawar and
+               Emanuela Merelli},
+  title     = {Towards Optimal Set-Disjointness and Set-Intersection Data Structures},
+  booktitle = {47th International Colloquium on Automata, Languages, and Programming,
+               {ICALP} 2020, July 8-11, 2020, Saarbr{\"{u}}cken, Germany (Virtual
+               Conference)},
+  series    = {LIPIcs},
+  volume    = {168},
+  pages     = {74:1--74:16},
+  publisher = {Schloss Dagstuhl - Leibniz-Zentrum f{\"{u}}r Informatik},
+  year      = {2020},
+  url       = {https://doi.org/10.4230/LIPIcs.ICALP.2020.74},
+  doi       = {10.4230/LIPIcs.ICALP.2020.74},
+  timestamp = {Tue, 30 Jun 2020 17:15:44 +0200},
+  biburl    = {https://dblp.org/rec/conf/icalp/KopelowitzW20.bib},
+  bibsource = {dblp computer science bibliography, https://dblp.org}
+}

macros.tex

@@ -328,3 +328,7 @@
 
 %%%Adding stuff below so that long chain of display equatoons can be split across pages
 \allowdisplaybreaks
+
+\newcommand{\eps}{\epsilon}
+\newcommand{\inparen}[1]{\left({#1}\right)}
+\newcommand{\inset}[1]{\left\{{#1}\right\}}

main.tex

@@ -176,7 +176,7 @@ sensitive=true
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \bibliographystyle{plain}
-\bibliography{aaron.bib}
+\bibliography{aaron,atri}
 
 
 

mult_distinct_p.tex

@@ -23,6 +23,14 @@ Given a positive integer $k$ and  an undirected graph $G$ with no self-loops or
 
 The above result means that we cannot hope to count the number of $k$-matchings in $G=(V,E)$ in time $f(k)\cdot |V|^{O(1)}$ for any function $f$. In fact, all known algorithms to solve this problem take time $|V|^{\Omega(k)}$.
 
+Our hardness result in Section~\ref{sec:single-p} is based on the following conjectured hardness result:
+\begin{hypo}
+\label{conj:graph}
+There exists a constant $\eps_0>0$ such that given an undirected graph $G=(V,E)$, computing exactly the values $\numocc{G}{\tri}$, $\numocc{G}{\threepath}$ and $\numocc{G}{\threedis}$ cannot be done in time $o\inparen{|E|^{1+\eps_0}}$.
+\end{hypo}
+Based on the so called {\em Triangle detection hypothesis} (cf.~\cite{triang-hard}), which states that detection whether $G$ has a triangle or not takes time $\Omega\inparen{|E|^{4/3}}$, implies that in Conjecture~\ref{conj:graph} we can take $\eps_0\ge \frac 13$.
+\AR{Need to add something about 3-paths and 3-matchings as well.}
+
 To prove our hardness result, consider a graph $G(V, E)$, where $|E| = \numedge$, $|V| = \numvar$, and $i, j \in [\numvar]$.
 
 Consider the query $\poly_{G}(\vct{X}) = q_E(X_1,\ldots, X_\numvar) = \sum\limits_{(i, j) \in E} X_i \cdot X_j$.