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- W2952954087 abstract "Given a family of 3-graphs $F$, we define its codegree threshold $mathrm{coex}(n, F)$ to be the largest number $d=d(n)$ such that there exists an $n$-vertex 3-graph in which every pair of vertices is contained in at least $d$ 3-edges but which contains no member of $F$ as a subgraph. Let $F_{3,2}$ be the 3-graph on ${a,b,c,d,e}$ with 3-edges ${abc,abd,abe,cde}$. In this paper, we give two proofs that $mathrm{coex}(n, F_{3,2})= n/3 +o(n)$, the first by a direct combinatorial argument and the second via a flag algebra computation. Information extracted from the latter proof is then used to obtain a stability result, from which in turn we derive the exact codegree threshold for all sufficiently large $n$: $mathrm{coex}(n, F_{3,2})= lfloor n/3 rfloor -1$ if $n$ is congruent to $1$ modulo $3$, and $lfloor n/3 rfloor$ otherwise. In addition we determine the set of codegree-extremal configurations." @default.
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- W2952954087 date "2013-06-29" @default.
- W2952954087 modified "2023-10-17" @default.
- W2952954087 title "The codegree threshold for 3-graphs with independent neighbourhoods" @default.
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