Abstract
Let
$$\mathcal {T}$$
T
be a collection of 3-element subsets
$$S$$
S
of
$$\{1,\dots ,n\}$$
{
1
,
⋯
,
n
}
with the property that if
$$i<j<k$$
i
<
j
<
k
and
$$a<b<c$$
a
<
b
<
c
are two 3-element subsets in
$$S$$
S
, then there exists an integer sequence
$$x_1<x_2<\cdots <x_n$$
x
1
<
x
2
<
⋯
<
x
n
such that
$$x_i,x_j,x_k$$
x
i
,
x
j
,
x
k
and
$$x_a,x_b,x_c$$
x
a
,
x
b
,
x
c
are arithmetic progressions. We determine the number of such collections
$$\mathcal {T}$$
T
and the number of them of maximum size. These results confirm two conjectures of Noam Elkies.
