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Thank you to everybody for the great suggestions. I have decided what to do (see below) and I'll let you know how it goes after the picnic next week. I was inspired by the audience participation asp …

Here are a few examples from graph-Ramsey theory. In the first pair of examples, the Ramsey version and density version are essentially as far apart as one can get. In the last two pairs of examples …

For the minimum question, this is answered in Theorem 4 of Erdős, Paul; Füredi, Zoltán; Tuza, Zsolt, Saturated (r)-uniform hypergraphs, Discrete Math. 98, No. 2, 95-104 (1991). ZBL0766.05060. The answ …

I need to give a math talk to a group of undergraduates. I am asking for advice because this talk will take place at a department picnic and there will be no blackboard or projector. I would like to …

Given a graph $H$, let $R_k(H)$ be the smallest integer $N$ such that in every $k$-coloring of the edges $K_N$ there is a monochromatic copy of $H$ (in other words, $R_k(H)$ is the ordinary $k$-color …

This is related to a previous question that I asked. The degeneracy of a graph $G$, denoted $\mathrm{degen}(G)$, is given by $\max\{\delta(H): H\subseteq G\}$. It is well known that for all graphs $G …

I just stumbled across this question and see that it is five years old, but since I know the reference I thought I might as well share it. This threshold is determined in the paper "Local Connectivit …

Now to add to Jon's comment. Just take the case $t=2$ and suppose for simplicity that $n$ is divisible by 4. I'm guessing that $ex_2(n,K_{1,2,2})=\frac{n^2}{4}+\frac{n}{2}$ by taking a complete bala …

These examples are symmetric digraphs, i.e. graphs. For graphs, the Nash-Williams conjecture just becomes Chvatal's theorem (If $G$ is a graph on $n\geq 3$ vertices with degree sequence $d_1\leq d_2\ …

After reading relep's answer I revisited the problem and came up with a different fairly simple proof for Question 1, but before I get to that, I recently found that this problem has a long history al …

My first thought for the case where $|V|\leq \aleph_0$ is that surely the Rado graph can be constructed as an induced subgraph of $([\omega]^{\omega}, E)$ (since the Rado graph contains a copy of ever …

This is too long for a comment, but it's really just a modification of John Tuwim's answer. By using the reductions discussed in the original post, the proof becomes even simpler and it shows that $$ …

I have a two part question: Is there a simple proof that every strongly connected tournament $T$ on $n\geq 4$ vertices contains distinct $u,v\in V(T)$ such that $T-u$ and $T-v$ are strongly connected …

It seems to me that the answer should be yes, but my naive attempts to come up with an example have failed. Just to clarify, by finite hyperbolic geometry I mean a finite set of points and lines such …

This isn't a full answer, but too long for a comment. I suppose it comes closest to trying to answer Question 3 or the general question of whether the hypercube design can be improved. Definition Giv …