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Consider the bipartite graph whose left vertices are goods and right vertices are buyers. Draw an edge between each good $i$ and buyer $m$ with weight $p_{mi}$. Now, you want to find a max-weight bipa …

Intuitively, it should approach zero fast as soon as $|\mathcal S_{\mathsf{big}}|$ is at all smaller than $n$ (even like $n/2$) because virtually all subsets $\mathcal S_{\mathsf{small}}$ will contain …

I wonder if this perhaps-naive approach can help you. Let's shift by the mean, so consider $X = \sum_{j=1}^n X_j$ where each $X_j = 0.5$ or $-0.5$ independently with one-half probability each. So $\ma …

Fleshing out Boris Bukh's idea. We can draw $\pi$ by first sending $1$ uniformly to somewhere in $\{1,\dots,n\}$, then sending $2$ uniformly to the remaining $n-1$ spots, and so on. Consider a small …

Throw $m$ balls into $n$ bins independently, each ball selecting a bin from the distribution $A \in \Delta_n$. This question is about lower-bounding the max-loaded bin. Background. In this MO answer I …

If you only want upper bounds, there is a nice approach based on collisions. (Proving that the max-loaded bin is not too small w.h.p. seems to need completely different techniques.) Define a $k$-way …

The contrapositive of PH is: If you put at most one pigeon per hole, then you have at most $n$ pigeons. Letting $a = (1,\dots,1)$ and $b_i$ be the number of pigeons in hole $i$ with $b_i \in \{0,1\}$, …

I am looking for a construction that can be stated as the following coding problem: a binary code with good distance ($d = \Omega(n)$ where codeword length is $n$) that "resists local decoding" in the …

Since this is a well-studied problem, people will be a bit skeptical that you have found a significant improvement that is correct. I would try to look for someone you can contact via email and ask to …

(Preface: This may be a naive or easy question for experts....) Consider an infinite tree, rooted on the left, where each node has two children; the number of nodes at each level (distance from the r …

The answer is indeed asymptotic to $4n/\log(n)$, but I don't know of an elementary or easy construction for this upper bound. I believe simple probabilistic-method type constructions do not work. Thi …

You can use a "dynamic programming" solution. For anyone unfamiliar with this terms, the basic idea is that there are an exponential in $n,m$ number of possible paths, so it takes too long to enumerat …

Here's one. You can think of the graph construction process as gradually building a set $S$ of vertices that have been touched so far, beginning with a random two vertices. Let $S_k$ be the set of the …

The theoretical Computer Science perspective may be useful here. At least, TCS has developed a rigorous and precise sense in which what I'll call "discrete Brouwer" (François' "Approximate Fixed Point …

I think this is a natural question but am not sure where to find resources. Consider the possible multisets arising from choosing $n$ times an item from one of $k$ categories. We can represent one su …