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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.

14 votes
1 answer
560 views

Is there an elementary proof of a better result for the finite guessing-box puzzle?

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 …
Joel David Hamkins's user avatar
14 votes

Size of maximal intersecting families

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 …
Joel David Hamkins's user avatar
6 votes
Accepted

Is every finite poset a subset of a finite complemented distributive lattice?

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 $p$. …
Joel David Hamkins's user avatar
2 votes
Accepted

Can $\omega$ be parity-separated with finitely many bijections?

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 …
Joel David Hamkins's user avatar
8 votes

Seymour's second neighborhood conjecture for infinite graphs

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 …
Joel David Hamkins's user avatar
9 votes

Sunflowers in maximal almost disjoint families

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 …
Joel David Hamkins's user avatar
3 votes

Combinatorially defined effectively closed set

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 …
Joel David Hamkins's user avatar
3 votes
Accepted

Is following function a metric on the set of isomorphism classes of graphs with countably ma...

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 …
Joel David Hamkins's user avatar
2 votes

Infima and suprema in the "transfer" function ordering

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 …
Joel David Hamkins's user avatar
2 votes
Accepted

Can the union of difference sets in towers equal $\omega$?

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 …
Joel David Hamkins's user avatar
16 votes

Alice and Bob playing on a circle

For even $n$, I claim that nobody has a winning strategy, and therefore both players have drawing strategies. To see this, observe first that by the fundamental theorem of finite games, we know that …
Joel David Hamkins's user avatar
9 votes
3 answers
1k views

The Sudoku game: Solver-Spoiler variation

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 …
Joel David Hamkins's user avatar
19 votes
3 answers
1k views

The arithmetic progression game and its variations: can you find optimal play?

Consider the arithmetic progression game, a two-player game of perfect information, in which the players take turns playing natural numbers, or finite sets of natural numbers, all distinct, and the fi …
Joel David Hamkins's user avatar
58 votes
Accepted

Does knight behave like a king in his infinite odyssey?

Consider the following open knight's tour on a $5\times 5$ board, starting at position $1$ and then touring the $5\times 5$ board in the indicated move order. The final position is $25$, from which th …
Joel David Hamkins's user avatar
134 votes

What is a chess piece mathematically?

In terms of mathematical analysis and combinatorial game theory, the essence of any game is captured by its game tree, the tree whose nodes represent the current game state, and to make a move in the …
Joel David Hamkins's user avatar

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