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Theorem: There is no such $E$. Claim: for each $a\in E$ there exists $b\in E$ with $|b\setminus a|\ge 2$. Proof of Claim: Let $a$ be a counterexample. Then all $b\ne a$ contain exactly one element eac …

Of course all subfamilies of $2^{[n/2-1]}$ are sparse. For any $k$, the number of union-closed families $\mathcal F\subseteq 2^{[k]}$ is $u(k)=2^{\binom{k}{\lfloor k/2\rfloor}(1+o(1))}$ (link), where …

Here is a counterexample of size 23. Let $m=6$ and let $$P=\{0,a_1,\dots,a_m,1\}\cup\{b_{ij}: 1\le i<j\le m\}$$ where $0<a_i<b_{jk}<1$ whenever $i$ is distinct from $j$ and $k$. The cardinality of $P$ …

Let $s$ consist of $2^{2^k}$ zeros, followed by the same number of ones, for increasing $k$: $$0^21^2\, 0^{16}1^{16}\, 0^{256}1^{256}\,0^{65536}1^{65536}\dots$$ Observe that all but finitely many bloc …

You can use this: if $u\ne x$, $$dist(u,x)=1+\min\{dist(w,x):uw\in E\}.$$

Let $A_t$ be the event that $S_1$ is a substring of $S_2$, $S_2=pS_1q$, where the length of $p$ is $t$. Then the probability of $\cup_t A_t$ can be found by inclusion-exclusion as $$\sum P(A_t)-\sum P …

A word is cubefree if it cannot be written as $xyyyz$ where $y$ has positive length. Let $h$ be the morphism from $\{0,1,2,3,4\}^*$ to $\{0,1\}^*$ given for words of length 1 as follows ($a\to h(a)$) …

Imagine that you lay out the N (0) and E (1) moves as follows ($n=4$ shown): $$0000$$ $$1111$$ As you go along the path, color $\color{red}{red}$ the ones you have used, so that after reading either 0 …

We may assume $n\le m$ throughout. Let $y>x>0$. There are only finitely many $n$ with $$ 1/n\ge x/2. $$ For each such $n$, the set of $1/n+1/m$ as $m$ varies is not dense in any subinterval of $(x,y …

This is studied in Reverse Mathematics as the Chain Antichain Principle (CAC) and it is observed that it follows from Ramsey's theorem.

It is true. Let $k=\binom{N}{N/2-c\sqrt N}$ and let $K$ be a randomly* selected subset of $k$ of size $k\lambda$. Then conditionally on $|X|>N/2-c\sqrt N$, the difference $|X|-(N/2-c\sqrt N)$ is unbou …

Euler in 1769 conjectured that for all integers n and k greater than 1, if the sum of k nth powers of positive integers is itself a nth power, then k is greater than or equal to n: $$a^n_1 + a^n_2 …

Let $C_k$ be the set of partitions of $n$ containing $k$. Following @MaxAlekseyev's point we have, $$P(n-1)+P(n-2)-P(n)=|C_1|+|C_2|-P(n),$$ $$=|C_1\cap C_2|+|C_1\cup C_2|-P(n)$$ This is the # of part …

For $n=1$ the probability that such $A$, $B$ exist is at least $$\frac6{\pi^2}\left(1+\frac18\right)=68\%$$ since it can happen at least in the following two disjoint ways: $a$ and $b$ coprime $a$, …

Many people may be interested in the asymptotics for $n=cm$ where $c$ is constant (say $c=2$). That is, how likely is a function from $2m$ to $m$ to be onto? I couldn't dig the answer out from some o …