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Extended comment, generalizing @IlyaBogdanov's comment about $2n-1$. Fix $n$ and let $$x_m = a(m, n) + n - m - 1.$$ Then $(x_m)$ obeys the similar recurrence $$ x_m = x_{m-1} + \gcd(x_{m-1}, n-m) - …

Just playing around with it: The RHS multiplied by $n$ is the same as $$2 \sum_{k=0}^{n-1} \binom{n+1}{k} \binom{n}{k} \binom{n+1}{k+2}.$$ Subtracting this from $n$ times the LHS gives $$\sum_{k=0}^{n …

The maximum is $cn^{4/3}$ for some constant $c$. We may as well assume all our points are on the unit sphere $S$. Let $P$ be some plane not containing the origin, which we might think of as being far …

Suppose we have some positive quantites $P$ and $Q$ which depend on some choices that we make, and we want to show that some choice makes the quotient $P/Q$ fall below some cool bound. One idea is to …

Suppose a group $G$ acts on a set $\Omega$. Call a subset $A \subset G$ regular (or sharply transitive or simply transitive or...) if for every two points $\omega_1, \omega_2 \in \Omega$ there is a un …

The difference cover problem has been better studied in the context of $\mathbf{Z}$. Redei, Renyi, and others in the 40s asked for the size of the smallest set $A$ such that $A-A$ covers $\{1,2,\dots, …

Claim 1: (same as my comment) Let $q_1, \dots, q_k > 1$ be prime powers. Then $G = C_{q_1} \times \cdots \times C_{q_k}$ is isomorphic to a subgroup of $S_n$ if and only if $q_1 + \cdots + q_k \leq n$ …

Repeating from the comments section: This (natural and beautiful) question was previously asked and answered on this site. See Collapsible group words. It also appeared recently on math.se. The questi …

Let $a +_K b$ denote Knuth addition. It is easy to check that $a +_K b \equiv a + b$ mod $2$ (in fact mod $4$), so the odd numbers are Knuth-sum-free. On the other hand, note that if $a +_K b = a +_K …

As noted by Seva, you're looking for Szemeredi's cube lemma, which is one of the simplest results of its kind. Let $S\subset\mathbf{F}_2^n$, and let $\|S\|_{U^k}^{2^k}$ denote the number of $e_0,e_1, …

Observe that, in isolation, (the indicator function of) an arithmetic progression of common difference $2$ and length $N > n$ maps to an arithmetic progression of common difference $2$ and length $N-n …

Naturally, this was also considered by Erdös and Graham. Graham mentions at the top of page 10 here for instance: http://www.math.ucsd.edu/~ronspubs/13_03_Egyptian.pdf. I'm not aware of any progress.

Is there any known improvement on the Kahn-Kalai-Linial inequality (on the influences of boolean functions) in the special case in which $f$ is the indicator function of an intersecting monotonic set …

Well, here's a combinatorial proof of the identity anyway (but not a direct proof of independence of $\ell$). Write $N = R - t$. Then the identity is $$\sum_{n=0}^N \binom{n + \ell}{n} \binom{R - n-\e …

Nice problem! I claim that $\limsup n^2 p(n) = \infty$. Suppose $k < n/2$ is such that $n-k$ is divisible by $L_k = \text{lcm}(1,2,\dots,k)$. Then if $\pi \in S_n$ has a cycle of length $n-k$ (this h …