  Young flattenings, introduced by Landsberg and Ottaviani, give determinantal
equations for secant varieties and their non-vanishing provides lower bounds
for border ranks of tensors and in particular polynomials. We study
monomial-optimal shapes for Young flattenings, which exhibit the limits of the
Young flattening method. In particular, they provide the best possible lower
bound for large classes of monomials including all monomials up to degree 6,
monomials in 3 variables, and any power of the product of variables. On the
other hand, for degree 7 and higher there are monomials for which no Young
flattening can give a lower bound that matches the conjecturally tight upper
bound of Landsberg and Teitler.
