Minimal hypergraph non-jumps
Benedict Randall Shaw

TL;DR
This paper develops a general framework for the Frankl-R"odl method to identify non-jumps in hypergraph densities, providing new smaller non-jumps and showing the method's limitations.
Contribution
It introduces a general setting for the Frankl-R"odl method and applies it to find new, smaller non-jumps in hypergraph densities, highlighting the method's bounds.
Findings
Identified new non-jumps smaller than previously known.
Demonstrated the current method's limitations in proving smaller non-jumps.
Abstract
An -uniform hypergraph, or -graph, has density . We say is a jump for -graphs if there is some constant such that, for each and , any sufficiently large -graph of density at least has a subgraph of order and density at least . For , all are jumps. For , Erd\H{o}s showed all are jumps, and conjectured all are jumps. Since then, a variety of non-jumps have been proved, using a method introduced by Frankl and R\"odl. Our aim in this paper is to provide a general setting for this method. As an application, we give several new non-jumps, which are smaller than any previously known. We also demonstrate that these are the smallest the current method can prove.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Point processes and geometric inequalities · Markov Chains and Monte Carlo Methods
