Low-degree monomial count for the cap-set bound ($m_n$)

The number $m_n$ of monomials in $n$ variables with each exponent at most two and total polynomial degree at most $\lfloor 2n/3\rfloor$ satisfiesFor each exponent vector $a$ counted on the left, $1\le 2^{2n/3-{\sum }_i{a}_i}$. Summing and then allowing all $a\in \{0,1,2{\}}^n$ gives $m_n\le 2^{2n/3}(1+1/2+1/4{)}^n$. The base has cube $343/16<27$, so it is strictly less than three. This elementary weighted count gives the exponential saving in the polynomial method in combinatorics without an unstated probability estimate.