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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

4 votes
1 answer
187 views

"JigSaw Puzzle" on Set Family II

I was asked to post a different question following a wording error on my previous question, so here it is. Given a set family $\mathcal{F}$ of $[n]$ (with certain additional properties), such that ev …
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2 votes
0 answers
39 views

Covering base sets $X$ with a subset family satisfying a "partial covering property"

Let $X$ be an $n$ element base set. Suppose I have a subset family $\mathscr{F} \subset 2^X$ satisfying the following property: (*) For any subset $Y \subset X$, we can find an element $F \in \mathscr …
abacaba's user avatar
  • 384
0 votes
1 answer
157 views

"JigSaw Puzzle" on Set Family

One of my research problem can be reduced to a question of the following form Given a set family $\mathcal{F}$ of $[n]$ , such that every element of $[n]$ lies in exactly $K$ sets in $\mathcal{F}$, c …
abacaba's user avatar
  • 384
2 votes

Does an $(x, bx)$-biregular graph always contain a $x$-regular bipartite subgraph?

Here is a heuristic reason why such a claim cannot hold. I cannot make the claim rigorous as I don't know enough about random biregular bipartite graphs. If the claim indeed holds, then for any $(x, b …
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  • 384
4 votes
0 answers
70 views

A question about the existence of surjective contractions

A few years ago I was doing some research in origami, and was motivated to as the following questions: Consider $\mathbb{R}^2$ with the Euclidean metric and Lebesgue measure. Does there exist a const …
abacaba's user avatar
  • 384
11 votes
0 answers
194 views

Number of triangle-free graphs with prescribed number of edges

This question is posted from StackExchange since it received no answer there. Let $f(n, e)$ be the number of triangle-free graphs on $n$ vertices and $e$ edges. From empirical evidence, I am motivated …
abacaba's user avatar
  • 384