Patrascu (STOC '10) reduces the 3SUM problem to
listing triangles in a graph. In the other direction, we
show that if one can solve 3SUM on a set of size $n$ in
time $n^{1+\e}$ then one can list $t$ triangles in a
graph with $m$ edges in time $\tilde
O(m^{1+\e}t^{1/3-\e/3})$. Our result builds on and
extends works by the Paghs (PODS '06) and by Vassilevska
and Williams (FOCS '10). We make our reductions
deterministic using tools from pseudorandomness.

We then re-execute both Patrascu's reduction
and ours for the variant 3XOR of 3SUM where integer
summation is replaced by bit-wise xor. As a corollary we
obtain that if 3XOR is solvable in linear time but
3SUM requires quadratic randomized time, or vice versa,
then the randomized time complexity of listing $m$
triangles in a graph with $m$ edges is $m^{4/3}$ up to a
factor $m^\alpha$ for any $\alpha > 0$.

We show that if one can solve 3SUM on a set of size $n$
in time $n^{1+\epsilon}$ then one can list $t$ triangles in a
graph with $m$ edges in time $\tilde
O(m^{1+\epsilon}t^{1/3+\epsilon'})$ for any $\epsilon' > 0$. This is a
reversal of Patrascu's reduction from 3SUM to
listing triangles (STOC '10).

We then re-execute both Patrascu's reduction
and our reversal for the variant 3XOR of 3SUM where
integer summation is replaced by bit-wise xor. As a
corollary we obtain that if 3XOR is solvable in linear
time but 3SUM requires quadratic randomized time, or vice
versa, then the randomized time complexity of listing $m$
triangles in a graph with $m$ edges is $m^{4/3}$ up to a
factor $m^\alpha$ for any $\alpha > 0$.

Our results are obtained building on and extending works
by the Paghs (PODS '06) and by Vassilevska and Williams
(FOCS '10).