This question is posted from StackExchange since it received no answer there.
Let $f(n, e)$ be the number of triangle-free graphs on $n$ vertices and $e$ edges. From empirical evidence, I am motivated to make the following conjecture.
Conjecture: $f(n,e)$ is unimodal in $e$. In other words, for each $n$ there exists a mode $m = m(n)$ such that $f(n, e) \leq f(n, e + 1)$ if $e < m$, and $f(n, e) \geq f(n, e + 1)$ if $e \geq m$.
I am wondering if this conjecture has been mentioned in literature before since it is natural for such sequences to display some unimodality.
Here's another way to phrase the question.
Conjecture: Let $G$ be the $3$-uniform linear hypergraph whose vertex set is $\binom{[n]}{2}$, and whose edge set is $\{\{i,j\},\{j, k\}, \{k, i\}\}$ for distinct $i,j,k\in [n]$. Then the independence polynomial $Z_G(x)$ is unimodal.
It is known that if we instead consider the $2$-uniform graph with the same vertex set and edge set $\{\{i,j\},\{j, k\}\}$, then $Z_G(x)$ is equal to the matching polynomial $m_G(x)$ of $K_n$, which is unimodal and log-concave.
