823 research outputs found

    Shadows and intersections: stability and new proofs

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    We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lov\'asz's theorem that answers a question of Frankl and Tokushige.Comment: 18 page

    Supersaturation and stability for forbidden subposet problems

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    We address a supersaturation problem in the context of forbidden subposets. A family F\mathcal{F} of sets is said to contain the poset PP if there is an injection i:P→Fi:P \rightarrow \mathcal{F} such that p≤Pqp \le_P q implies i(p)⊂i(q)i(p) \subset i (q). The poset on four elements a,b,c,da,b,c,d with a,b≤c,da,b \le c,d is called butterfly. The maximum size of a family F⊆2[n]\mathcal{F} \subseteq 2^{[n]} that does not contain a butterfly is Σ(n,2)=(n⌊n/2⌋)+(n⌊n/2⌋+1)\Sigma(n,2)=\binom{n}{\lfloor n/2 \rfloor}+\binom{n}{\lfloor n/2 \rfloor+1} as proved by De Bonis, Katona, and Swanepoel. We prove that if F⊆2[n]\mathcal{F} \subseteq 2^{[n]} contains Σ(n,2)+E\Sigma(n,2)+E sets, then it has to contain at least (1−o(1))E(⌈n/2⌉+1)(⌈n/2⌉2)(1-o(1))E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2} copies of the butterfly provided E≤2n1−εE\le 2^{n^{1-\varepsilon}} for some positive ε\varepsilon. We show by a construction that this is asymptotically tight and for small values of EE we show that the minimum number of butterflies contained in F\mathcal{F} is exactly E(⌈n/2⌉+1)(⌈n/2⌉2)E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}
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