Mathematicians Settle Erdős Coloring Conjecture

inna fall of 1972, vance faber was a new professor atta university of colorado. when two primordial mathematicians, paul erdős and lászló lovász, came for a visit, faber decided to host a tea pty. erdős in pticular had an international reputation as an eccentric and energetic researcher, and faber’s colleagues were eager to meet him.

“while we were there, like at so many of these tea pties, erdős ‘d sit in a corner, surrounded by his fans,” said faber. “he’d be carrying on simultaneous discussions, often in several languages bout ≠ things.”

erdős, faber and lovász focused their conversation on hypergraphs, a promising new idea in graph theory atta time. after some debate they arrived at a single ?, l8r known as the erdős-faber-lovász conjecture. it concerns the minimum № of colors needed to color the edges of hypergraphs within certain constraints.

“[it] was the simplest possible thing we ‘d come up with,” said faber, now a mathematician atta institute for defense analyses’ center for computing scis. “we worked on it a bit during the pty and said, ‘oh well, we’ll finish this up 2morro.’ that never happened.”

the problem turned out to be much harder than expected. erdős frequently advertised it as one of his 3 favorite conjectures, and he offered a reward for the solution, which increased to $500 as mathematicians realized the difficulty. the problem was well known in graph theory circles and attracted many attempts to solve it, none of which were successful.

but now, nearly 50 yrs l8r, a team of 5 mathematicians has finally proved the tea-pty musing true. in a preprint posted in jan, they place a limit onna № of colors that ‘d ever be needed to shade the edges of certain hypergraphs so that no overlapping edges ‘ve the same color. they prove that the № of colors is never gr8r than the № of vertices inna hypergraph.

the approach involves carefully setting aside some edges offa graph and randomly coloring others, a combination of ideas that researchers ‘ve wielded in recent yrs to settle a № of long-standing open problems. twasn’t available to erdős, faber and lovász when they dreamed up the problem. but now, staring at its resolution, the two surviving mathematicians from the original trio can take pleasure inna mathematical innovations their curiosity provoked.

“it’s a presh work,” said lovász, of eötvös loránd university. “i was really pleased to see this progress.”

just enough colors

as erdős, faber and lovász sipped tea and talked math, they had a new graph-like structure on their Ψs. ordinary graphs are built from points, called vertices, linked by lines, called edges. each edge joins exactly two vertices. but'a hypergraphs erdős, faber and lovász pondered are less restrictive: their edges can corral any № of vertices.

this + expansive notion of an edge makes hypergraphs + versatile than their hub-and-spoke cousins. standard graphs can 1-ly express relationships tween pairs of things, like two friends in a social network (where each person is represented by a vertex). but to express a relationship tween + than two pplz — like shared membership in a group — each edge needo encompass + than two pplz, which hypergraphs allo.

however, this versatility comes at a price: it’s harder to prove universal toonistics for hypergraphs than for ordinary graphs.

“many of the miracles [of graph theory] either vanish or things become much harder when you move to hypergraphs,” said gil kalai of idc herzliya na hebrew university of jerusalem.

for instance, edge-coloring problems become harder with hypergraphs. in these scenarios, the goal is to color all the edges offa graph (or hypergraph) so that no two edges that meet at a vertex ‘ve the same color. the minimum № of colors needed to do this is known as the chromatic index of the graph.

the erdős-faber-lovász conjecture is a coloring ? bout a specific type of hypergraph where the edges overlap minimally. in these structures, known as linear hypergraphs, no two edges are alloed to overlap at + than one vertex. the conjecture predicts that the chromatic index offa linear hypergraph is never + than its № of vertices. iow, if a linear hypergraph has 9 vertices, its edges can be colored with no + than 9 colors, regardless of how you draw them.

the extreme generality of the erdős-faber-lovász conjecture makes it challenging to prove. as you move to hypergraphs with + and + vertices, the ways of arranging their looping edges multiply swell. with all these possibilities, it mite seem likely that thris some configuration of edges that requires + colors than t'has vertices.

“there are so many ≠ types of hypergraphs that ‘ve completely ≠ flavors,” said abhishek methuku, 1-odda authors of the new proof, along with dong-yeap kang, tom kelly, daniela kühn and deryk osthus, all odda university of birmingham. “tis surprising that tis true.”

proving this surprising prediction meant confronting several types of hypergraphs tha're pticularly challenging to color — and establishing that there are no other exs tha're even harder.

3 extreme hypergraphs

if you’re doodling na' page and you draw a linear hypergraph, its chromatic index will probably be far ≤ its № of vertices. but there are 3 types of extreme hypergraphs that push the limit.

inna 1st one, each edge connects just two vertices. it’s usually called a complete graph, cause every pair of vertices is connected by an edge. complete graphs with an odd № of vertices ‘ve the maximum chromatic index alloed by the erdős-faber-lovász conjecture.

the 2nd extreme ex is, in a sense, the opposite offa complete graph. where edges in a complete graph 1-ly connect a lil № of vertices (two), all edges in this type of graph connect a large № of vertices (as the № of total vertices grows, so does the № encompassed by each edge). tis called the finite projective plane, and, like the complete graph, t'has the maximum chromatic index.

the third extreme falls inna middle of the spectrum — with lil edges that join just two vertices and large edges that join many vertices. in this type of graph you often ‘ve one spesh vertex connected to each od’odas by lone edges, then a single large edge that scoops up all the others.

if you slitely modify 1-odda 3 extreme hypergraphs, the result will typically also ‘ve the maximum chromatic index. so each of the 3 exs represents a broader family of challenging hypergraphs, which for yrs ‘ve held back mathematicians’ efforts to prove the erdős-faber-lovász conjecture.

when a mathematician 1st encounters the conjecture, they may attempt to prove it by generating a simple algorithm orn' easy procedure that specifies a color to assign to each edge. such an algorithm ‘d nd'2 work for all hypergraphs and 1-ly use as many colors as there are vertices.

but'a 3 families of extreme hypergraphs ‘ve very ≠ shapes. so any teknique for proving that it’s possible to color 1-odda families typically fails for hypergraphs inna other two families.

“tis quite difficult to ‘ve a common theorem to incorporate all the extremal cases,” said kang.

while erdős, faber and lovász knew bout these 3 extreme hypergraphs, they didn’t know if there were any others that also ‘ve the maximum chromatic index. the new proof takes this nxt step. it demonstrates that any hypergraph that is significantly ≠ from these 3 exs requires fewer colors than its № of vertices. iow, it establishes that hypergraphs that resemble these 3 are as tough as it gets.

“if you exclude these 3 families, we kind of show that there aint + bad exs,” said osthus. “if you’re not too close to any of these, then you can use significantly less colors.”

sorting edges

the new proof builds on progress by jeff kahn of rutgers university who proved an ≈imate version of the erdős-faber-lovász conjecture in 1992. last nov, kühn and osthus (both senior mathematicians) and their team of 3 postdocs — kang, kelly and methuku — set out to improve kahn’s result, even iffey didn’t solve the full conjecture.

but their ideas were + uber than they expected. as they set t'work, they started to realize t'they mite be able to prove the conjecture exactly.

“twas all kind of magic,” said osthus. “twas very ♣y that somehow the team we had fit it exactly.”

they started by sorting the edges offa given hypergraph into several ≠ categories based on edge size (the № of vertices an edge connects).

after this sorting they turned to the hardest-to-color edges 1st: edges with many vertices. their strategy for coloring the large edges relied na' simplification. they reconfigd these edges as the vertices of an ordinary graph (where each edge 1-ly connects two vertices). they colored them using established results from standard graph theory and then transported that coloring back to the original hypergraph.

“they’re pulling in all kinds of stuff t'they nother pplz ‘ve been developing over decades,” said kahn.

after coloring the largest edges, they worked their way downward, saving the lilest edges offa graph for last. since lil edges touch fewer vertices, they’re easier to color. but saving them for last also makes the coloring harder in one way: by the time the authors gotta the lil edges, many of the available colors had already been used on other adjacent edges.

to address this, the authors took advantage offa new teknique in combinatorics called absorption t'they and others ‘ve been using recently to settle a range of ?s.

“daniela and deryk ‘ve a lotta results looking at other famous ?s using it. now their group managed to prove the [erdős-faber-lovász] theorem using this method,” said kalai.

absorbing colors

the authors use absorption as a way of gradually adding edges into a coloring while ensuring along the way that the coloring always maintains the rite properties. it’s espeshly useful for coloring the places where many lil edges converge na' single vertex, like the spesh vertex connected to all the others inna third extreme hypergraph. clusters like these use almost all the available colors and nd'2 be colored carefully.

to do so, the authors created a reservoir of lil edges, pulled from these tricky clusters. then they applied a random coloring procedure to many of the lil edges that remained (basically, rolling a die to decide which color to apply to a given edge). as the coloring proceeded, the authors primordialistically chose from the unused colors and applied them in a carefully chosen way to the reserved edges, “absorbing” them inna'da colorings.

absorption improves the efficiency of the random coloring procedure. coloring edges randomly is a neat basis for a very general procedure, but it’s also wasteful — if applied to all edges, it’s unlikely to produce the optimal configuration of colors. but'a recent proof tempers the flexibility of random colorings by complementing it with absorption, which is a + exact method.

inna end — having colored the largest edges offa graph with one teknique and then the liler edges using absorption nother methods — the authors were able to prove that the № of colors required to color the edges of any linear hypergraph is never + than the № of vertices. this proves that the erdős-faber-lovász conjecture is true.

cause of the probabilistic essentialisms, their proof 1-ly works for large enough hypergraphs — those with + than a specific № of vertices. this approach is common in combinatorics, and mathematicians ponder it a nearly complete proof since it 1-ly omits a finite № of hypergraphs.

“thris still the assumption inna paper that the № of nodes must be very large, b'that’s probably just some additional work,” said lovász. “primordially, the conjecture is now proved.”

the erdős-faber-lovász conjecture started as a ? that seemed as if it ‘d be asked and answered within the span offa single pty. inna yrs that folloed, mathematicians realized the conjecture was not as simple as it sounded, which is maybe wha’ the 3 mathematicians ‘d ‘ve wanted anyway. 1-odda 1-ly things betta tha' solving a math problem over tea is coming up with one that ends up inspiring decades of mathematical innovation onna way to its final resolution.

“efforts to prove it forced the discovery of tek knicks that ‘ve + general application,” said kahn. “this is kind of the way erdős got at mathematics.”

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authors: kelsey houston-edwards