Non-jumps of hypergraphs
Vaughn Komorech
TL;DR
The paper addresses the non-jump problem in Tur\'an densities for $r$-uniform hypergraphs by developing a pattern-based framework. Building on the Frankl–Rödl method, it introduces $r$-patterns and their blowups, and uses Lagrangian optimization to reduce asymptotic densities to maximizing $r!p_P(x)$ over the standard simplex. A key mechanism, the FR$_v(P)$ construction, yields a concrete non-jump criterion: if $\lambda(\mathrm{FR}_v(P))=\lambda(P)<1$ with a positive weight on $v$, then $r!\lambda(P)$ is not a jump, enabling systematic non-jump discoveries. The authors apply this method to specific $3$-patterns, obtaining the classic non-jump $\frac{3}{4}$ for small patterns and proving new non-jumps, notably $\frac{64}{81}$ for $r=3$, along with a parametric family of non-jumps indexed by $n$. This pattern-based approach extends the toolkit for probing the jump/non-jump landscape beyond previously known intervals and offers a scalable route to new non-jump results in hypergraph theory.
Abstract
A density $α\in [0, 1)$ is a jump for $r$ if there is some $c >0$ such that there does not exist a family of $r$-uniform hypergraphs $\mathcal{F}$ with Turán density $π(\mathcal{F})$ in $(α, α+ c)$. Erdös conjectured that all $α\in [0, 1)$ are jumps for any $r$. This was disproven by Frankl and Rödl when they provided examples of non-jumps. In this paper, we provide a method for finding non-jumps for $r = 3$ using patterns. As a direct consequence, we find a few more examples of non-jumps for $r = 3$.