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The infinitary guessing-box puzzle is amazing — see here. In the basic form, the Guessing-box Hall has infinitely many wooden boxes, each containing a real number, and there are 100 mathematicians wh …

The answer is yes. Consider first for simplicity the case where $X$ is countably infinite. If $\mathcal{S}$ is a maximal intersecting family, then I claim that $\mathcal{S}$ must contain a set with in …

Complementing Andres's excellent answer, let me simply try to help build your intuition for elementary submodels. The basic situation is just like the familiar fact that if you have finitely many grou …

As Sam mentioned in the comments, the answer to question 1 is yes. One can map every condition $p$ in the partial order to the lower cone, the set $S_p=\{q\in P\mid q\leq p\}$ of conditions below $p$. …

No, because if you have $n$ functions, then the number of possible parity patterns to be exhibited by a number with respect to them is $2^n$. So by the pigeon-hole principle there must be infinitely m …

Allow me to make an observation concerning what I find to be an interesting angle on the question in the context without the axiom of choice, where there are competing conceptions of what it means to …

Apart from your specific example, the idea of truth-by-accident has been studied in the context of formal first-order languages, which includes the language of graph theory, and in his dissertation, K …

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each c …

Consider the Sudoku Solver-Spoiler game, a natural variation of the Sudoku game recently appearing in the question Who wins two-player Sudoku? posted by user PyRulez. In that game, the players attempt …

It is consistent with ZFC that the answer to your question is no. Specifically, I claim, if we assume the continuum hypothesis, then the answer is no, not even for sunflowers of size $n=3$. Theorem. A …

A partial order $\mathbb{B}$ is universal for a class $\cal{P}$ of partial orders if every order in $\cal{P}$ embeds order-preservingly into $\mathbb{B}$. For example, every partial order $\langle\ma …

The answer is yes. Let me first describe a general method. Fix any c.e. computably inseparable pair $A$ and $B$. These are computably enumerable sets having no computable separation. There are diver …

To prove that this is a metric, consider the following theorem. Theorem. If the second player can survive for $n$ steps in the $(\Gamma_1,\Gamma_2)$ game, and for $m$ steps in the $(\Gamma_2,\Gamma_3 …

My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers. The ques …

Here is a counterexample showing that the quotient of $\text{Fct}(X,Y)$ is not necessarily a lattice. Let $X=\{0,1,2\}$ and let $Y=\{0,1\}$. Let $g(0)=g(1)=0$ and $g(2)=1$, while $f(0)=0$ and $f(1)=f …