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revised On independent sets of graph
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revised On independent sets of graph
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revised On the notion of positive rank
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revised On independent sets of graph
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comment On independent sets of graph
I do not think they are equivalent definitions $\frac{n}{\alpha(G)}<n\ll\binom{n}{\alpha(G)}$.
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revised Number of ways to extend almost independent sets of graph
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comment On independent sets of graph
So if $\max(\alpha(G),N(G))<|V|^{\frac{1}{cr}}$, then we could hope $|V|^{\frac{r-1}{cr}}<|V|^{\frac{cr-1}{cr}}\leq M(G)$ is possible. It seems that if $c\gg1$, an example of graph family is feasible.
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revised Is following matrix sets convex?
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revised Is following matrix sets convex?
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revised Is following matrix sets convex?
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revised Is following matrix sets convex?
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comment Number of ways to extend almost independent sets of graph
@YuvalFilmus Just curious if $N(G)$ been studied anywhere before in literature? It seems easy to find a family with $$\alpha(G)\leq|V|^{\frac{1}{r+3}}\leq|V|^{1-\frac{1}{r+3}}\leq M(G)$$ $$N(G)\leq|V|^{\frac{1}{r+3}}\leq|V|^{1-\frac{1}{r+3}}\leq M(G)$$ Controlling both $\alpha(G),N(G)$ seems tough.
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revised Is following matrix sets convex?
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asked Is following matrix sets convex?
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revised Number of ways to extend almost independent sets of graph
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revised Number of ways to extend almost independent sets of graph
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comment On correctness of induction proof
@AvZ comments on my update query?
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revised Probabilistic statement on matrix ranks
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