Three layer $Q_2$-free families in the Boolean lattice

We prove that the largest $Q_{2}$ -free family of subsets of $[n]$ which contains sets of at most three different sizes has at most $(3 + 2\sqrt {3})N/3

o(N) \approx 2.1547N + o(N) $m e m b e r s, w h e r e$ N = {n \choose {\lfloor n/2 \rfloor}} $. T h i s i m p r o v e s a n e a r l i e r b o u n d o f$ 2.207N + o(N)$ by Axenovich, Manske, and Martin.

We prove that the largest $Q_{2}$ -free family of subsets of $[n]$ which contains sets of at most three different sizes has at most $(3 + 2\sqrt {3})N/3

o(N) \approx 2.1547N + o(N) $m e m b e r s, w h e r e$ N = {n \choose {\lfloor n/2 \rfloor}} $. T h i s i m p r o v e s a n e a r l i e r b o u n d o f$ 2.207N + o(N)$ by Axenovich, Manske, and Martin.

Three layer $Q_{2}$ -free families in the Boolean lattice

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Three layer $Q_{2}$ -free families in the Boolean lattice

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Three layer Q2Q2​-free families in the Boolean lattice

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Three layer Q2Q2​-free families in the Boolean lattice

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Three layer $Q_{2}$ -free families in the Boolean lattice

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Three layer $Q_{2}$ -free families in the Boolean lattice

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AI智能校验