Scientists recently announced progress on an elusive problem from 1960 known as the Sunflower Conjecture. While it doesn’t involve any plant biology, it does deal with objects called “mathematical sunflowers,” which are arguably cooler than the garden staple. The latest news isn’t a complete solution to the longstanding conjecture, but it is a significant step forward after decades of relative inactivity.

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The story of this math problem—like so many others—starts with a Hungarian mathematician named Paul Erdős, who holds the all-time record for the most publications in mathematics. He’s basically the Kevin Bacon of math, where your degree of separation from publishing with Paul Erdős is called your Erdős Number .

Erdős frequently collaborated with British mathematician Richard Rado, and in 1960, they wrote what is known as the Sunflower Lemma. The term “lemma” essentially means “mini theorem,” which is something that’s been proven true, but isn’t major enough to be outright called a theorem. Their lemma specified a certain estimate, and they hypothesized that the estimate could be improved—a claim that became known as the Sunflower Conjecture.

What’s in the Lemma?

Dave Linkletter

Here’s the idea behind mathematical sunflowers. Take some sets of numbers, and look at the ones that are common between sets. If every pair of sets has the same numbers in common, we think of those as the central disk of the sunflower. The remaining parts of each set are called the petals of the sunflower, and so the number of sets is the number of petals.

For example, take the sets {1, 2, 3, 4}, {1, 3, 5, 7}, and {1, 3, 9, 27}. They form a sunflower with three petals, with a central disk (mathematically nicknamed the Kernel of the sunflower) of {1,3}. Notice that 1 and 3 are in each of the three sets, and no numbers appear in exactly two out of the three sets.

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The definition is flexible, which helps the outcome be so robust. Technically, a sunflower’s kernel can have as few as 0 elements. So the sets {1, 2, 3}, {4, 5, 6}, and {7, 8, 9} form a sunflower with three petals and a kernel with no elements. This means mathematical sunflowers can be all petals, no disk. Unlike real sunflowers.

The sets of the sunflower can be different sizes, just like real-life sunflowers don’t have all identical petals, so that one actually makes sense. Simply keep track of the maximum size of the sets; the authors of the new paper, including a mathematician and three computer scientists, notate it “w.” For the number of petals in the flower, they use the letter “r”. Now you’re ready for the big fact in its pure form.

The Sunflower Lemma says that if you have at least w!(r-1)w sets, each with up to w numbers, then there must be a sunflower with r petals. That factorial tells you the numbers get huge quickly, but for computers’ purposes, it’s a beautifully succinct estimate, and was a triumph for Erdős and Rado as they worked on this emerging subject together.

They also predicted their lemma could be improved—specifically, that the w!(r-1)w could be upgraded to some c(r)w, a constant to the power of w. That prediction was the Sunflower Conjecture.

The new result isn’t a full answer to Erdős and Rado’s conjecture, aiming to write that bound as some constant to the power w. But—brace yourself for the longform—it improves that estimate to (log w)w(r log log w)o(w). Much messier than the old estimate, right? Yes, but it’s also a significant upgrade.

Different functions heading toward infinity might do so at different speeds, in the sense that one function is always larger than the other as they grow infinitely. Mathematicians have very precise terms for comparing these things, but all you need is the natural intuition of “growing faster.”

The original estimate w!(r-1)w is a function that grows very quickly. The desired form c(r)w

would grow profoundly slower. And the new estimate is right in the middle, a magnitude of improvement from the original, but also a magnitude away from the goal.

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This is nevertheless exciting news for mathematicians and computer scientists, particularly since so little has happened since the problem was posed. After 1960, it went decades with no notable progress, until one publication in 1997, and another in 2019. As always with math breakthroughs, we now get to see if further results will quickly follow, perhaps leading to the full solution of the conjecture.

One follow-up paper has already been written, though it does not get any closer to a complete answer; it’s more of a cleaned-up rewrite. But it does show scientists are already at work trying to utilize the newest breakthrough.

On the math side, this news falls into the subject of Combinatorics, the study that includes large finite calculations of combinations and situations like this. The most basic combinatorics questions sound like “how many 5-card poker hands have two pairs?”, while the most advanced combinatorics questions sound like the Sunflower Conjecture.

For computer scientists, this is about computational complexity. They like to know how long it will take a computer to do a task, given knowledge about the size of the task. Finding a mathematical sunflower amongst a plethora of sets starts as a purely computational exercise, but once we know something about the speed of solving it, we can use it to solve other problems you might not expect.