# Tag Info

Accepted

### Does the hypergraph of subgroups determine a group?

In the comments to the question, I notice something which might be an error, or at least is an incomplete response. It is pointed out in the comments that there exist nonisomorphic groups with ...
• 10.4k

### Does the hypergraph of subgroups determine a group?

For large prime $p$, there are uncountably many non-isomorphic Tarski monsters of exponent $p$. For these groups $G$, the subgroups lattice consists of basically a partition of $G\smallsetminus\{1\}$...
• 55.2k
Accepted

### Singular cardinal $\kappa$ with projective plane such that all edges have cardinality $<\kappa$

The answer is no. Let $\kappa$ be any infinite cardinal, regular or singular, and assume for a contradiction that there is a set $E\subseteq\mathcal P(\kappa)$ satisfying your conditions. I will call ...
• 9,221
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• 11.9k
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### Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor

As explained in a previous MO question, there is no unique generalization of the eigenvalue of an $n\times n$ matrix to an $n\times n\times n$ tensor. One approach is to construct a higher-order ...
• 160k
Accepted

### Is it true that any $3$-uniform hypergraph that is not $k$-colorable must have $\Omega(k^3)$ edges?

The following paper of Alon shows that the quantity you're after, $m(k)$, the minimum number of edges of a $3$-uniform hypergraph which is not $k$-colourable, is indeed $\asymp k^3$. More precisely, ...
• 6,353

### Non-isomorphic hypergraphs on $\omega$

Others have already answered, but I think the following counting argument is worth pointing out: there are $2^{2^{\aleph_0}}$ hypergraphs on $\omega$ (since a hypergraph on $\omega$ is just a ...
• 25.4k
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### Are hypergraphs $H=(V,E)$ with $|E|=|V|$ $2$-colorable?

This is essentially done by the Bernstein set construction: if one has $\kappa$ many sets each of size $\kappa$, then order them into ordinal $\kappa$ and recursively choose 2 points from each, so ...
• 7,144
Accepted

### Intersecting subsets of $\{1,\ldots,n\}$

No, there isn't. This is essentially the dual version of the De Bruijn-Erdos theorem if the elements of $\mathcal C$ are the points, and the elements from $\{1,\ldots,n\}$ are the lines. The original ...
• 17.9k

• 3,626
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### A generalization of Erdős-Ko-Rado theorem

The case $s=1$ is Erdős hypergraph mathcing conjecture from Paul Erdős (1965). A problem on independent $r$-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 8 (1965), 93–95. users.renyi.hu/~...
• 2,926
Accepted

### Maximum intersecting set families of $\{1,\ldots,n\}$

The intersecting family in your example has $\binom{n-1}{\lfloor\frac{n-1}{2}\rfloor}$ members by Sperner's theorem. An example that achieves a larger value would be to take all the subsets of $[n]$ ...
• 83.2k

### Does the hypergraph of subgroups determine a group?

As @Keith Kearnes says, the negative answer ought to be somewhere in Roland Schmidt's book. Unless I'm mistaken, it suffices to find two non isomorphic groups with isomorphic coset lattices. Indeed, ...
• 3,227
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• 761