Non-jumping densities of 3-uniform hypergraphs

Summary · qwen2.5:32b

The paper introduces a method to identify non-jumping densities specifically for 3-uniform hypergraphs by utilizing patterns, contradicting Erdős's conjecture that all densities are jumps. This matters as it advances the understanding of Turán densities in hypergraph theory. The authors provide new examples of non-jumps for $r = 3$ as a result of their method.

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In this paper, a method for finding non-jumps in the context of 3-uniform hypergraphs is presented, shedding light on number theory and combinatorics.

Excerpt

arXiv:2511.07715v2 Announce Type: replace Abstract: A density $\alpha\in [0, 1)$ is a jump for $r$ if there is some $c >0$ such that there does not exist a family of $r$-uniform hypergraphs with Tur\'an density in $(\alpha, \alpha + c)$. Erd\"os conjectured that all $\alpha\in [0, 1)$ are jumps for any $r$. This was disproven by Frankl and R\"odl 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 $r = 3$.