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The answer is no, even if you consider only injective functions. Consider any infinite cardinal $\kappa$, and let $X=\omega\times\kappa$ be the disjoint union of $\kappa$ many copies of $\omega$. Le …

Every family $\mathcal{F}$ containing $\omega$ itself as a member is trivially maximal, since no infinite set is almost disjoint from $\omega$. But the family could otherwise consist of an almost disj …

I claim that Q can have no winning strategy, even in the fully general game where there is no restriction on the complexity of sets to be played. To see this, suppose that we have a strategy $\sigma$ …

Nice question. The answer is no, not necessarily. Theorem. There is a graph $G$ such that there is no minimal vertex covering of it by maximal cliques. Indeed, in every vertex covering $\cal C$ of $G …

For the title question, the answer is no, when d is less than c, because you could map your set into the 2-valued functions and map the complement to the rest. So the image would not be dominating.

Your requirements are inconsistent; there is no insane family. Suppose towards contradiction that we have an insane family $\mathcal{B}=\{B_\alpha\mid\alpha\lt\kappa\}$, witnessed by tower $\langle A …

The general fact is that every mathematical structure of size $\kappa$, in a language of size at most $\kappa$, can be coded as a (connected, undirected, simple) graph of size $\kappa$. What I mean is …

Gerhard has pointed out that your sharing-a-point graph is not universal for uncountable graphs, since any uncountable collection of functions on $\omega$ must have many of them sharing a point. So th …

Yes, this can happen, and indeed it happens in a subforest of any given Souslin tree. Let's start with an illustrative case. Sometimes people consider Suslin trees that are not necessarily normal, an …

Take any increasing tower, but then modify it by adding all the numbers below $n$ to $A_n$, when $n$ is even, and removing them when $n$ is odd. This is a finite change to each set in the tower, and s …

This is not a full answer, but I claim that the restriction to regular towers is not relevant. Theorem. The cardinal $\hat t$ is also the supremum of all tower sizes, not just the regular towers. $$ …

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 …

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree wit …

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 …

Here is some general background information. The relevant search phrases for this topic are generalized cardinal invariants or generalized cardinal characteristics, and the topic has a growing literat …