A well-known conjecture in mathematics, known as the Sunflower Conjecture, was introduced and investigated by Erdős and Rado in 1960. Since then, many mathematicians have worked to understand its mysteries. However, the conjecture remains unsolved to this day.

A sunflower brings to mind a particular image: Petals are neatly arranged, fanning out from a circular mass. We can simplify this imagery by drawing a series of loops, each representing one petal.

In this simplistic visualization, a sunflower can be made up of many such loops in such a way that you are reminded of a child’s drawing.

Now, let’s imagine that each of these loops contains some pebbles. If a pebble is contained in more than one of a sunflower’s loops, it will automatically be in its center, assuming that the loops intersect at the center.

In this way, we have effectively described a sunflower as a collection of loops.

However, loops don’t always form a sunflower. Loops can be arranged in such a way that they do not form a sunflower. For example, if there are three loops but only two of them intersect, then those three loops do not form a sunflower.

Something interesting happens here. Imagine a collection of loops and assume that each of them contains, say, seven pebbles. If I have many such loops, will a sunflower always appear? It turns out that if this collection has 130 loops, then no matter how you arrange these loops, you can always find a sunflower with at least three petals. And if you increase the number of these loops further, you will eventually find sunflowers with even larger numbers of petals.

In 1960, Erdős and Rado proved that this always happens.

In this so-called Sunflower Conjecture, they conjectured that the growth rates of these numbers, such as 130 mentioned above, are at most exponential functions on the number of pebbles in each loop. For example, if the Sunflower Conjecture were true for sunflowers with three petals, then we can find some number relating to the aforementioned growth rate, perhaps an extremely large number such as 1,000,000,000, that will enable us to make interesting conclusions.

If I have, say, 1,000,000,000^2000 (means to the power of 2000) loops with 2000 pebbles in each loop, then among all of those loops, I can always find at least three of them that form a sunflower with at least three petals. I can also make the same conclusion when there are 1,000,000,000^7 loops with seven pebbles in each loop. In the above, I stated that 130 such loops are enough, but there are no problems here as 130 is much smaller than 1,000,000,000^7.

What makes this nice is not that 1,000,000,000^7 is much larger than 130, it is that I can find a single number, such as 1,000,000,000, which captures all pebble sizes (from one to infinity) for sunflowers with three petals.

Fundamentally, we are interested in determining the smallest number that captures all the pebble sizes and whether such a number exists.

Deepening this mystery is that no one knows if such numbers even exist.

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Let us take a deeper look into some of the numbers involved.

Recall our earlier example involving 130, with seven pebbles in each loop.

If we increase the number of pebbles in each loop to eight, then it turns out that you will need 258 loops to have the same result. That is, if there are 258 loops with eight pebbles in each loop, you can always find three loops that form a sunflower with three petals.

By how much did the number grow when transitioning from seven pebbles to eight pebbles? We calculate that 258/130 is about 1.98, which is nearly two. What happens if we go further? Further increasing the number of pebbles per group to nine gives 514 as our answer. Comparing this number with the previous one, we obtain 514/258, which is about 1.99.

The sequence 1.98, 1.99 may seem to approach 2, but that turns out to be false. If we were to extend the above sequence of ratios by making further calculations, we would eventually begin to find that many of these ratios are at least 3.16. This is a consequence of a result by Abott, Hansen, and Sauer. This is a far cry from numbers like 1,000,000,000. But no one knows whether these ratios will stop growing.

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Our discussion of sunflowers is indicative of the fact that there are very difficult questions relating to this that can be asked. The Sunflower Conjecture may open the door to many mysteries further ahead.