The worst-case scenario was proposed by Kahn and Kalai. The gap between the expectation threshold and the true threshold will never be greater than a logarithmic factor according to their eponymous conjecture. The general answer to the central question in random graphs is given by the conjecture.
That is a simple case. More than random graphs are the subject of the conjecture. It holds for random sequence of numbers, for generalizations of graphs called hypergraphs, and for even broader types of systems. The statement was written in terms of abstract sets. A random graph can be thought of as a random subset of the set of all possible edges, but there are many other objects that fall within the conjecture.
Shachar Lovett is a theoretical computer scientist at the University of California, San Diego. It would make it easier to calculate thresholds for different properties.
It seemed crazy to many mathematicians that so many seemingly unrelated problems could be solved. Kahn and Kalai didn't try to prove it. They were trying to find a counterexample. They figured they were going to discover one of the many settings they could explore.
Kalai said that the story evolved in a very different way than they had expected.
The breakthrough on the seemingly unrelated problem that would lead to the new proof of the Kahn-Kalai conjecture was the beginning of the methods that would eventually lead to the new proof. The story begins with a question posed by the mathematicians in 1960. Collections of sets can be constructed in ways that look like the petals of a sunflower.
A team of people came very close to a solution of the sunflower problem. The work seemed completely different from the Kahn-Kalai conjecture. We weren't aware of these other questions. We cared about the flowers.
When they set out to prove a more relaxed version of the Kahn-Kalai conjecture, they ended up linking the two. Their proof was published in the Annals of Mathematics. The weaker version was formulated by the French mathematician, and it had a different way of taking a weighted average. The revised definition gives you more wiggle room to work with things.
The team realized they could apply the techniques from the sunflower result to the Talagrand conjecture.
The mathematicians worked on the problem iteratively. They wanted to show that a random set would contain a Hamiltonian cycle. Park said that they perform some sort of random process iteratively and that they chose it in pieces.
The team came up with the idea of spread to do this. If Hamiltonian cycles are spread out, it means that there are different edges to each cycle.
The approach was possible only because of a critical equivalence: Spread could be quantified in a way that related directly to the fractional expectation threshold. The mathematicians were able to rewrite the Talagrand conjecture in terms of spread.
The proof of the weak conjecture was enough to settle a lot of threshold-related problems. It was suggested that the values of the fractional and original expectation thresholds were basically the same. The only way to prove the Kahn-Kalai conjecture was to prove that equivalence.
That isn't what happened. While other mathematicians tried to follow the road map, Park and Pham found a new approach. I wasn't expecting that at all.
It's one of the nice things about mathematics, he said.
Park and Pham didn't intend to tackle the original conjecture at first. Park said that she never imagined that she would prove it.
It wasn't in our mind at all.
When they were struck with a revelation, they were working on another conjecture. The picture that we have here, the ideas that we have, it somehow seemed like it was more powerful than we thought.
They figured out how to make the proof work after a sleepless night in March.
The normal expectation threshold has no relationship with spread. Give you a starting point by spreading it. The starting point of the non-fractional conjecture disappears if you go to the original one.
What do you do?
They thought about the problem in terms of a mathematical object called a cover. A cover is a collection of sets, where every object with a certain property contains one of those sets. The collection of all edges is a possible cover for all Hamiltonian cycles. One of those edges will be in every Hamiltonian cycle.
The Kahn-Kalai conjecture was rewritten in a way that made use of covers. There are constraints on what the probability of a weighted coin landing on heads should be in order to guarantee that a random graph or set contains some property. The expectation threshold for the property has to be at least a logarithmic factor. The weighted coin's probability is lower if the property is not likely to emerge.
When a small cover can be constructed for a subset of structures, it means that their contribution to the expectation threshold is small. The expectation threshold is calculated by taking a weighted average of all possible structures of a given type. If a random set is not likely to contain one target structure, there must be a small cover for all of them. The majority of their proof was devoted to that small cover.
They used a similar process to the one used in the previous results and introduced a very clever counting argument.
Their proof is very simple. The basic idea we developed and the ideas from other papers are added to a new twist.
I can't explain it, but it's true, he said.
Just as the fractional result implies a cornucopia of related conjectures, the Kahn-Kalai conjecture implies a cornucopia of related conjectures. Noga Alon said that this is a powerful proof technique that will probably lead to new things.
Park and Pham are applying their method to other problems. They are interested in getting a more precise understanding of the gap between expectation and real thresholds. They have shown that the gap is a logarithmic factor, but sometimes it is smaller. There is no broader mechanism for determining when each of these scenarios might be true, and mathematicians have to work it out case by case. With this efficient technique, we can hopefully be much more precise in pinning down these thresholds.
Park said that the proof could have other consequences.
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