I have funding!
Specifically, I will be working, if things go to plan, with Imre Leader on The Neighbourhood Conjecture. I am getting 1/5 or so of the money I'd get on an internship, and I may have a high probability of proving absolutely nothing [original], but oh well!
There are many [equivalent, which I shall prove for myself at an early stage] statements of the open problem, but it basically comes up in the topic of Hypergraph Games.
The idea of such a game, in its fully finite form, is quite simple - you have a set, and you have subsets of this set which are called "winning lines". Two players take turns claiming the points of the set, and the first player to claim an entire winning line wins. An alternative version of the game has the first player trying to win this way, and the second player only wins if when all the points in the set are claimed the first player has not won. This is called "maker-breaker".
An example of such a game is playing Noughts and Crosses on a 3-by-3 square board - the base set is the 3-by-3 grid, the winning lines are the eight three-in-a-row possibilities. We know this is a draw for normal play, but a win for first player under "maker-breaker" - try it out! Another example is noughts and crosses on a 3-by-3-by-3 board, which is a straightforward first-player win.
The easy way to state the problem is as follows: we look at games where winning lines are of fixed length, such as noughts and crosses (3), Gomoku (5), or three-dimensional four-in-a-row (4). If I am given a fixed length k for these lines, I want to consider games where the first player wins, and I want to simplify these in a very specific sense - I want to minimise the highest degree of any point involved, where the degree of a point is how many lines cross at it. Going back to the examples from earlier, 2D Noughts and Crosses has max degree 4, 3D has max degree 13, Gomoku has max degree 20, and 3D 4-in-a-row has some respectable number like 10. The current known lower bound is linear (by Hall polygamy) and the current known upper bound is exponential (by complete binary tree), so there's some room for improvement.
Incidentally, I'm already in the habit of talking about nhoods, even orally.
There are many [equivalent, which I shall prove for myself at an early stage] statements of the open problem, but it basically comes up in the topic of Hypergraph Games.
The idea of such a game, in its fully finite form, is quite simple - you have a set, and you have subsets of this set which are called "winning lines". Two players take turns claiming the points of the set, and the first player to claim an entire winning line wins. An alternative version of the game has the first player trying to win this way, and the second player only wins if when all the points in the set are claimed the first player has not won. This is called "maker-breaker".
An example of such a game is playing Noughts and Crosses on a 3-by-3 square board - the base set is the 3-by-3 grid, the winning lines are the eight three-in-a-row possibilities. We know this is a draw for normal play, but a win for first player under "maker-breaker" - try it out! Another example is noughts and crosses on a 3-by-3-by-3 board, which is a straightforward first-player win.
The easy way to state the problem is as follows: we look at games where winning lines are of fixed length, such as noughts and crosses (3), Gomoku (5), or three-dimensional four-in-a-row (4). If I am given a fixed length k for these lines, I want to consider games where the first player wins, and I want to simplify these in a very specific sense - I want to minimise the highest degree of any point involved, where the degree of a point is how many lines cross at it. Going back to the examples from earlier, 2D Noughts and Crosses has max degree 4, 3D has max degree 13, Gomoku has max degree 20, and 3D 4-in-a-row has some respectable number like 10. The current known lower bound is linear (by Hall polygamy) and the current known upper bound is exponential (by complete binary tree), so there's some room for improvement.
Incidentally, I'm already in the habit of talking about nhoods, even orally.