Tree suspensions and transfer functions for single degree Turán spectra

Abstract

For integers $1\le \ell <k$, let ${\Pi }_{\ell }^k$ denote the single-forbidden $\ell$-degree Turán spectrum of $k$-uniform hypergraphs. We introduce transfer functions for this spectrum: explicit functions $f$ such that, for every $F$, there is another single $k$-graph $F^{\ast }$ with ${\pi }_{\ell }(F^{\ast })=f({\pi }_{\ell }(F))$. This gives a mechanism for producing new single-forbidden densities while retaining full control of the resulting value. Our transfer functions are realized by a new family of suspension-type operations, called tree suspensions. From these operations we obtain three explicit maps: one acting on ${\Pi }_{\ell }^k$ for every $1\le \ell <k$, a second acting when $\ell \ge k/2$, and a third acting in the ordinary Turán case $\ell =1$. The common feature is a robust tree structure which gives the lower bound by a two-part construction and, in the regimes above, admits a matching embedding or Lagrangian upper bound. As a first application, the universal transfer function propagates accumulation points. Using the recent zero-accumulation results for $\ell \ge 2$ together with the ordinary Turán accumulation result of Conlon and Schülke, we prove that ${\Pi }_{\ell }^k$ has infinitely many accumulation points for every $k\ge 3$ and every $1\le \ell <k$. This recovers, in particular, the known infinitude of accumulation points in the ordinary and codegree spectra. As a second application, combining two independent transfer functions forces algebraic degrees to grow. For every $k\ge 3$ and every $\ell \in \{1,\lceil k/2\rceil ,\dots ,k-2\}$, the spectrum ${\Pi }_{\ell }^k$ contains algebraic numbers of arbitrarily large degree over $\mathbb{Q}$. Thus the arithmetic complexity previously known for finite forbidden families already occurs in the single-forbidden spectrum, both for ordinary Turán density and for a broad range of degree Turán densities.