Is your heart hiding from your fire?
Nov. 25th, 2025 05:27 pmI wrote about John von Neumann for the Free Press
Nov. 25th, 2025 09:25 pmThey asked me to do this piece as part of their series on great American immigrants. (Sorry, but it looks like this is paywalled for non-subscribers; I can’t see it, at any rate.)
It was fun to write, especially to get a chance to push back on the idea of von Neumann as a space alien who humans could barely comprehend. The real guy was fully human and much more interesting. There was a part I wanted to put in about how von Neumann’s late essay “Can We Survive Technology?” has a lot in common with the introduction to the Communist Manifesto, despite von Neumann’s vigorous anti-Communism. But I ran out of room and time. Maybe later!
Netflix’s Assassin’s Creed Series Snags Eden Actor Toby Wallace in Leading Role
Nov. 25th, 2025 08:41 pmNetflix’s Assassin’s Creed Series Snags Eden Actor Toby Wallace in Leading Role
Published on November 25, 2025
Photo: Bryan Berlin via Wikimedia Commons/Ubisoft
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Wednesday Season 3 Cast Adds Eva Green as Morticia’s Sister, Ophelia Frump
Nov. 25th, 2025 07:48 pmWednesday Season 3 Cast Adds Eva Green as Morticia’s Sister, Ophelia Frump
Published on November 25, 2025
Screenshot: Showtime
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emotional support coding
Nov. 25th, 2025 01:43 pm
I have Forth (programming language - see e.g. Leo Brodie's Starting Forth) running on this smol M5stack Cardputer v.1.1 (ESP32-S3) courtesy of ryu10's M5CardForth, which is also faster than my spending the next decade teaching myself ESP32-S3 assembler. :)
Next step: write a very smol choose-your-own-adventure-style text adventure in Forth.
Next step after that: ???
Next step after that: Considering porting either the Shuos Academy text adventure WIP [1] or Winterstrike (originally written for Failbetter Games for StoryNexus, which will be sunsetted by Jan 2026) to M5CardForth for the CardPuter because I am a TROLL. It could be a dumbass household game experience. :) :)
Heck, I could port some version of turnabout's fair prey or The Amiable Planet (Twine) to this! I love the thought of making TINY parser IF / text adventures for this smol device.
(All of these are my games. I give myself permission?!)
[1] I was writing/coding this for Choice of Games but we mutually agreed to cancel the contract because I was flooded out that year and it was no longer a doable workload alongside...finding new housing etc. I still have like 60% of the codebase already written in ChoiceScript and outline, though! I'd have to refactor but hell, I'd have to refactor anything. I can pretend it's pseudocode. :)
(I need a break from the current schoolwork, what can I say.)
Here Are All the Genre Movies Premiering in December!
Nov. 25th, 2025 07:00 pmThere is a lot of entertainment out there these days, and a lot of fantasy, sci-fi, and horror titles to parse through. So we’re rounding up the genre movies coming out each month.
Several horror movies hit theaters this month, including a sequel to Five Nights at Freddy’s and a reboot of an 80s slasher classic. The newest Avatar (the blue people, not the cartoon) movie also comes out.
Troll 2 — on Netflix December 1
What’s better than one giant ancient troll awakening in the Scandinavian wilderness and wreaking havoc upon the landscape? Two giant ancient trolls. Netflix’s original Norwegian monster epic became the most popular non-English language film when it released on the streaming service in 2021. The main cast returns for the sequel.
Five Nights at Freddy’s 2 — in theaters December 5
Based on the popular horror video game franchise of the same name, Five Nights at Freddy’s 2 takes place one year after the events of the first film. In the first movie, security guard Mike (Josh Hutcherson) spent five nights running from a group of haunted animatronics at an abandoned pizzeria. But his sister befriended the animatronics, so in this new movie she sneaks right back in. Five Nights at Freddy’s 2 will dive deeper into the origin of the pizzeria.
Dust Bunny — in theaters December 12
Mads Mikkelsen stars as a hitman for hire, who is asked by his young neighbor Aurora to kill the monster that lives under her bed. Aurora is convinced that the monster has eaten her family, but the hitman is pretty sure that her parents were killed by some rivals who actually meant to take him out. Dust Bunny comes from Hannibal and Pushing Daisies showrunner Bryan Fuller.
Silent Night, Deadly Night — in theaters December 12
A remake of the 1984 horror film of the same name, Silent Night, Deadly Night follows Billy Campbell, a murderer on who goes on a Christmas killing spree while dressed as Santa Claus. The original 80s movie spawned five installments. There was already a reboot in 2012, but it didn’t have much commercial success.
Resurrection — in select theaters December 12
In a future where humanity has lost the ability to dream, a woman dives into the dreams of a strange inhuman creature. She uses her ability to perceive illusions to figure out the creature’s visions of Chinese history. This surreal Chinese-language sci-fi drama is broken into six chapters, each representing one of the five senses (plus “the mind”).
Avatar: Fire and Ash — in theaters December 19
Return to the world of Pandora in the newest installment of James Cameron’s sweeping sci-fi epic. Last time, the Sully family learned the way of water by spending time with an aquatic Na’vi clan. This time, they go to the Fire Nation—wait, wrong Avatar. Ahem. This time, they visit an aggressive tribe of Na’vi known as the Ash people, who live alongside a volcano.
The SpongeBob Movie: Search for SquarePants — in theaters December 19
This is the fourth theatrical SpongeBob movie, following 2020’s Sponge on the Run. In this one, the ineffably cheerful sponge wants to prove his bravery to his miserly boss Mr. Krabs. So he travels to the deepest darkest depths of the ocean, in search of the Flying Dutchman.
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The post Here Are All the Genre Movies Premiering in December! appeared first on Reactor.
Here Are All the Genre TV Premieres Airing in December!
Nov. 25th, 2025 07:00 pmThere is a lot of entertainment out there these days, and a lot of fantasy, sci-fi, and horror titles to parse through. So we’re rounding up the genre shows coming out each month.
December might not have a lot of new genre shows coming out, but the ones that are premiering are all heavy-hitters. There are some highly anticipated shows returning, like Percy Jackson and the Olympians and Fallout. And Stranger Things ends, at long last! Yes, it also ended last month, but that was just part one of the ending. This time it’s for real!
Percy Jackson and the Olympians — Disney+ (December 10)
(Season 2) The second season of Disney’s Percy Jackson show adapts the second book in Rick Riordan’s Greek mythology middle grade series. Percy sets out on a quest to rescue his best friend, Grover the satyr, and also heal the protective pine tree that guards Camp Half-Blood. Along with Annabeth, daughter of Athena; Tyson, Percy’s newly discovered half-brother; and Clarisse, daughter of Ares, Percy journeys to the dangerous Sea of Monsters to recover the legendary Golden Fleece.
Tomb Raider: The Legend of Lara Croft — Netflix (December 11)
(Season 2) Netflix’s animated Tomb Raider show concludes with this last season. It takes place after the Tomb Raider Survivor trilogy and serves to bridge that timeline with the original games. Hayley Atwell voices adventuring archaeologist Lara Croft. In the first season, Lara ventured to a mythological Chinese mountain range in search of a goddess.
Fallout — Prime Video (December 17)
(Season 2) It’s a good month for video game adaptations! After an apocalyptic nuclear war, the Earth has become a wasteland and most survivors live in underground bunkers. A young woman named Lucy leaves her home and looks for her father in the remnants of Los Angeles. In the second season, she ventures to New Vegas — a location in popular Fallout spinoff, Fallout: New Vegas.
Stranger Things — Netflix (December 25 and 31)
(Season 5, Parts 2 & 3) Finally! The last season of Stranger Things — which premieres on the Wednesday before Thanksgiving — concludes. And don’t get too comfortable on Christmas Day, because the finale actually comes out on New Year’s Eve. The fifth and final season takes place a year after the fourth one. And the core cast wants to finally put an end to the creatures from the Upside Down, but their plan gets complicated when the U.S. military arrives.
The Copenhagen Test — Peacock (December 27)
Simu Liu stars in this sci-fi spy thriller, where he plays an intelligence analyst who learns his brain has been hacked. He must figure out who hacked into his brain, while proving his loyalty and competence to his agency. But he must do all that while the hackers have access to his every move and conversation.
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The post Here Are All the Genre TV Premieres Airing in December! appeared first on Reactor.
Mike Flanagan’s Exorcist Movie Is Still Happening & Scarlett Johansson Is Starring
Nov. 25th, 2025 06:53 pmMike Flanagan’s Exorcist Movie Is Still Happening & Scarlett Johansson Is Starring
Published on November 25, 2025
Screenshot: Universal Pictures
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Wake Up Dead Man Is a Miraculous Film
Nov. 25th, 2025 06:30 pmWake Up Dead Man Is a Miraculous Film
Published on November 25, 2025
Credit: Netflix
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Under the Sea: The Deep Range by Arthur C. Clarke
Nov. 25th, 2025 06:00 pmUnder the Sea: The Deep Range by Arthur C. Clarke
Published on November 25, 2025
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Two Things
Nov. 25th, 2025 08:36 am2. Mom gave me a ball of really pretty white yarn, which is wool with a bit of a halo, and maybe sport weight? Could be finger weight? Unsure? It's large enough to make a scarf, and I'm wondering if anyone can suggest a pattern that's: a) VERY EASY, b) maybe has a little bit of lace? I can do very basic lace, as long as it doesn't have too many steps.
Must Read Short Speculative Fiction: October 2025
Nov. 25th, 2025 04:30 pmMust Read Short Speculative Fiction: October 2025
Published on November 25, 2025
No themes for this spotlight, only vibes. I’m too tired for themes. My brain is goo and my energy level is negative 62. October sucked the life outta me, y’all. These are the ten science fiction, fantasy, and horror stories I read last month that managed to break through the noise.
“Affinity Gradient” by Miah O’Malley
“It had been two years since Ori’s burial, and her tree was thriving.” Dr Kar Reul is studying changes in a tree after her human love is buried in its roots. Most trees bound up with a human, their “anomalies” fade over time, but Ori’s seem to be getting stronger. Kar doesn’t know if she’s imagining these changes or if it’s really Ori in some new form trying to send a message. A compelling story about grief and loss. (Phano—October 2025; issue 10)
“Ghosts of Summer” by Catherine Tavares
Tavares jumps around in time with her story about two people summoning ghosts as a sort of supernatural air conditioner during the heat of summer. It starts off rather playful, just two people playing around with necromancy. The ghosts are not bound by the linearity of time, which leads to a shocking revelation. The narrative is little vignettes from different days over the course of a long summer, including their temperatures, so that by the time the temperature dips, you know something very bad has happened. (Apex Magazine—October 2025; issue 151)
“Knife Plus” by Tracie McBride
The first of two stories on this list with an inanimate thing as the protagonist. In this case, it is, of course, a knife. “Knife does not exist on its own. It is always Knife plus. Knife in forge. Knife on bench. Knife in hand. Knife in flesh.” Things take a, ahem, sharp turn toward the end of this flash fiction story when Knife realizes it isn’t just for butchering meat. The story cuts to the bone, figuratively and literally. (Fantasy Magazine—Autumn 2025; issue 98)
“Mother Tongue” by Pooja Joshi
I’ve read a lot of short stories about language loss, but this is the first where the language itself is the narrator. Here, a language brought to a new land slumbers and wakes as the few remaining speakers pick it up on and off through their lives. It’s a bittersweet look at the cultural costs of colonization, assimilation, and immigration. But Joshi also uses the words “slumber” and “sleep,” reminding us that even though our cultures can be lost, sometimes we can regain them, even if only in fragments. (Augur—October 2025; issue 8.2)
“Phantom View” by John Wiswell
The narrator, the son of a man on hospice, discovers a vaguely human-shaped “rusty orange-and-black blurry streak” that is visible only through a digital screen. While he tries to balance his dying father’s medical care and his own needs as a disabled person with shifting mobility issues, he finds a strange sort of comfort in the presence of this entity. He reaches out to the being, and the response is not what he expects. The entity felt a little like an analogy to a neurodivergent person who was non-verbal, someone whose perception of the world and ways of interacting with others differs from what is more common, but that also doesn’t make their needs or ways of expression any less valid. It’s always nice to have a story about the complexities of being disabled run through the speculative lens. (Reactor—October 22, 2025)
“The Pretendian” by Jason Pearce
This was so interesting! The protagonist of this story, a man calling himself Leroy Whiskeyjack, claims to be Lakota. But it’s not that simple. I’ll let you discover for yourself what the twist is on this story. All I’ll say is that when I finished I was thinking a lot about identity, culture, and cultural appropriation. This is The Deadlands, so get ready for blood and bones. (The Deadlands—Fall 2025; issue 40)
“Singularities” by Cressida Roe
This is a story about stories. It’s also a clever use of the multiverse trope. A man lives his life non-linearly. Sometimes he meets a woman, sometimes he doesn’t. Sometimes he rescues her, sometimes he doesn’t. Sometimes they lie in bed together, sometimes he’s alone. But always their stories circle back to each other. Will he ever be able to grab a hold of her and not let go? Which version of their endless lives is the real one? Does it matter? (Kaleidotrope—Autumn 2025)
“Soul Food” by C. M. Harmon
Family recipes aren’t just about the food. They carry our history and heritage, our stories and dreams. In Harmon’s piece, the protagonist summons the Crossroads Man in a last-ditch attempt to save their grandmother’s life. The price? A prized family recipe for candied yams. But it’s more than just some side dish. “No one in the family knows where the recipe came from. My grandma insisted it was old when her grandma passed it to her. And not too far past that, our family history drops off into the jagged-toothed maw that was slavery. Where some families have black and white photos and ship manifests, we have that recipe.” Giving up the yams means losing much of what the protagonist is trying to save. What will they choose? (FIYAH—Winter 2025; issue 36)
“The Superposition of Ramen” by Jed Looker
“For now, let us simply speak from your subjective standpoint. It is sometime in the early 21st century and you are reading what we understand to be a highly regarded science-fiction and fantasy magazine.” Okay, lol, you got my attention. The penultimate On Spec before it shutters forever is full of excellent stories, but this was my favorite. I will tell you nothing about it at all except it involves alternate timelines, aliens, and the life-altering choice between “a French bistro and a Japanese noodle house.” A funny, clever tale. (On Spec—October 2025; issue 133)
“A Taxonomy of Extinct and Extant Birds of the Twenty-First Century” by A.P. Golub
I love an unusual narrative format, and this piece is a good example of why. It’s structured as little snippets on various birds, and written in second person POV. Each segment tells the reader a little about the “you” in this story, about their meet-cute, marriage, and that person’s eventual illness. And because it’s a Reckoning joint, there’s a strong throughline of environmental commentary. (Reckoning—Fall 2025)[end-mark]
The post Must Read Short Speculative Fiction: October 2025 appeared first on Reactor.
Reading The Wheel of Time:The Gathering Storm (Part 9)
Nov. 25th, 2025 04:00 pmReading The Wheel of Time:The Gathering Storm (Part 9)
Published on November 25, 2025
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Consider Setting Your Space Operas on Saturn
Nov. 25th, 2025 03:00 pmConsider Setting Your Space Operas on Saturn
Published on November 25, 2025
Credit: GSFC/NASA
Credit: GSFC/NASA
It’s fashionable amongst persons of a certain age to moan about how the future has not lived up to expectations, how children are rude and everyone is writing a book, or how goods and services provided today are inferior to those provided forty years ago1. However, there is one aspect of our future that everyone I queried agreed has delivered beyond expectations (at least, once it became clear that was the most efficient way to end the conversation). That’s the state of planetary science.
There was a lamentable period between July 1965 (when Mariner 4 arrived at Mars) and March 1979 (when Voyager 1 arrived at Jupiter) in which the impression given by our space probes was of a dead, boring Solar System. The Moon was an airless dead rock, Mars was a nearly airless dead rock, Venus was an overheated dead rock, Mercury admittedly had a fascinating spin-orbit resonance but was still a dead rock.
Furthermore, the impression given wasn’t just of a dead Solar System, but a static death. Terms like “primordial” were thrown around. The Solar System wasn’t just dead—it had been dead and unchanging for billions of years. I can tell you, this was a major bummer for anyone who was, as I was, a kid in the 1960s and 1970s.
Once the images of Jupiter’s moons began trickling in, at a hilariously low baud rate, that changed. Jupiter’s moons were clearly dynamic worlds. Europa, at least, seemed to have a subsurface ocean. Our understanding of the Solar System was transformed, a process that is ongoing.
A convenient example of the pace of revelation would be John Varley’s 1979 Titan, whose plot is set in motion by the discovery in 2025 of a twelfth Saturnian moon. Unfortunately for Varley, the twelfth Saturnian moon is not a vast alien habitat suitable for thrilling adventures, as we learned when it was discovered just a year later, in 1980, along with two more moons. Even more moons followed.
In March 2025, the number of moons known to be orbiting Saturn jumped from 1462—already far more than Jupiter’s 97—to 274, thanks to the discovery of 128 previously undetected small moons. I am going to go out on a limb here and speculate that there are probably a metric whackload of small bodies that have yet to be spotted. Further, given that Saturn already has so many more known moons than Jupiter (even though Jupiter is much closer to us, which makes spotting Jovian moons easier) Saturn will probably be able to keep its lead3.
So, what does that matter to hard-working science fiction fans and writers? Aside from the sheer awesomeness of Saturn’s moon system, Saturn, or at least its moons, have tremendous potential, from a space opera perspective.
As you may recall from this semi-recent essay, space operas seem to require a multiplicity of worlds and a certain grandeur of scale. However, as I pointed out in that essay, given judicious worldbuilding choices, the Solar System can provide that scale. Saturn alone can deliver that scale.
Some folks will point out that those new moons are fairly small; most of them are barely over 3+ km diameters. That’s quite true. However, even a dinky moon is large in human terms. If my slide rule does not betray me, a 3 km diameter ice moon might mass about 12 billion tonnes, or fifteen New York Cities. Put together, the newly discovered moons probably mass about two thousand New York Cities.
Many space opera worlds give the impression of being not much larger than a Paramount backlot. The area of fifteen New York Cities combined is generally held to be larger than a Paramount backlot. Even ignoring the large moons (and why would we?), there’s enough easily accessible material in the Saturn system to justify a vast constellation of space cities4, each one of a size around which authors can easily wrap their minds.
Rather conveniently for your budding space opera author, many Saturnian moons abound in organic material, which is to say, the stuff from which living organisms are made5. Many protagonists are living organisms, so it’s pretty handy to have organic matter from which to construct them. In fact, there are speculations that moons such as Enceladus and Titan might already have life-bearing oceans; science is still out on that.
Saturn’s orbital dynamics offer intriguing plot potential. First, getting from one moon to another moon demands only modest delta vees and for most of the major moons, modest travel times.
Second, because it is so far from the Sun, Saturn is isolated in terms of delta vee and travel time from the other planets. Getting around within Saturn’s moon system will always be inherently faster and easier than travelling to Jupiter or to the inner system. Therefore, Saturn’s penumbra is a natural region.
Third, while the largest of Saturn’s moons orbit share a common plane and direction, Saturn’s smaller, irregular, moons include a number of dynamic families, moons whose paths around Saturn are similar to the other members of the family but different from other moons. These may be remnants of bodies whose capture by Saturn was kinetically exciting. Examples include the Inuit group, the Gallic group, and various subgroups of the Norse group, such as the Phoebe and the Kari. Again, because travel within families will be easier than travel between families, those families represent natural divisions.
Granted, transportation barriers aren’t the only factor defining borders. Just ask Ireland or Korea.
Admittedly, there is one resource in which the Saturn moon system is deficient6: heavy elements. Materials readily on hand around Saturn are mostly lighter elements and their compounds. Still, Saturn seems adept at capturing passing objects and it would not be surprising if a handful of M-type asteroids had been captured. Scarce resources are, of course, the stuff of which plot-driving conflicts are made.
I would argue that Saturn’s moons appear to offer every detail essential to space opera, with the added benefit of allowing authors to delay writing by investing time in calculating orbital parameters. Plus, Saturn has those beautiful rings, a spectacle no other planet can match!7 So, if you’re considering writing a space opera, consider setting it around Saturn.
- For an example that has personal relevance, old time appliances were often as durable as T-34s and about as energy efficient.
︎ - Not counting the ring particles as individual moons, or the count would be considerably larger. Speaking of the rings, because they orbit within the Roche limit of a gas giant, their orbital velocity is high and therefore retrieving material from the rings would be difficult. Well, except for groups living in the rings themselves. I don’t know that a habitat in the middle of a region filled with small, extremely fast-moving objects would last all that long. Yeah, the rings are mostly orbiting in the same direction at roughly similar speeds, but it still seems like it would be like living in a revolving cement truck drum filled with gravel. Plus, Saturn’s main rings have “an energetic particle and gamma ray photon radiation environment” which may not appeal. We probably don’t have to worry about corporations strip-mining the rings. Worst-case scenario, artists might squabble over what colour to paint the rings.
︎ - Jupiter has the lead with respect to giant moons, four to Saturn’s one. However, most of the Jovian giant moons are within Jupiter’s powerful Van Allen belts, where an unprotected human could receive a lethal radiation dose in the time it took to drink a cup of tea. Although since the moons also don’t have atmospheres as such, an unprotected human would be doing very well not to die in the first minute or so anyway. Not to mention their tea would boil away in the vacuum.
︎ - Without necessarily being fast. Saturn’s farthest moons have orbital periods in excess of four years. Assuming minimum energy transfer orbits, travel time could be measured in years.
︎ - And which can also be used to make plastic, which I suspect will dominate building materials.
︎ - Two, including the current lack of space habitats. Three, if fusion isn’t an option. The downside of being so far from the Sun is that solar power will be that much more difficult, although it should still be workable.
︎ - The rings, being thin, are essentially invisible to co-planar moons. This constitutes full disclosure for real estate regulatory purposes.
︎
The post Consider Setting Your Space Operas on Saturn appeared first on Reactor.
“dearly beloved, we are gathered here today to get through this thing called ‘life’”
Nov. 25th, 2025 07:34 am---L.
Subject quote from Let’s Go Crazy, Prince and the Revolution.
Sphere packing, cap set, and the Turán problem are all the same thing
Nov. 25th, 2025 04:37 amOK, they are not really the same thing. But I got you reading, right?
Here’s the sense in which they’re all the same thing. Let G be a group and H < G a subgroup. Let m > 1 be an integer. Write Orb_m for the set of orbits of G by simultaneous left multiplication on (G/H)^m, and f for the natural map from (G/H)^m to Orb_m. Let R be a subset of Orb. We say a subset S of (G.H) is an R-set if f(S^m) lies in R.
Master question: How large can an R-set be?
A lot of natural and popular questions are of this form, and I would argue that the unpopular questions of this form are also pretty natural! Perhaps this observation has been made before, but it’s new to me.
(Remark that you should only read if you’re pedantic: if R doesn’t contain m-tuples with repeated elements, then S^m can’t map to R, because S^m does have elements like that. So silently, in the examples below, I will mostly mean S choose m rather than S^m. I don’t want to commit to that forever, so if you really want it to be S^m, you can deal with that by appopriately adding some degenerate orbits to R.)
Some examples:
Sphere packing: G is AO_n, the affine orthogonal group or group of rigid motions, and H is O_n, so G/H is R^n. Take m=2. Then Orb is identified with the nonnegative real numbers, and f takes a pair of points to the distance between them. Let R be the interval [d,D]. Then an R-set is a set of points in R^n such that the distance between any two is at most D (i.e. we’re contained in some fixed big sphere) and at least d (the points don’t get too close together.)
(If having to choose an upper bound D annoys you, feel free to take G = O_n and H = O_{n-1}; then Orb is a finite interval [0,r] and you can just take R to be [d,r].)
Cap set: G is the group of affine linear transformations of F_p^n, and H is GL_n(F_p), so G/H = F_p^n. Take m=3 again. Orb is whatever it is, but one of the orbits is the set of (x,y,z) such that x,y,z are distinct and x-2y+z=0. Call this orbit, oh, I dunno, 3AP. Let R = Orb – 3AP. Then an R-set is a cap set, and indeed we want to know how big an R-set can be.
Turán problem: G is S_n and H is S_2 x S_{n-2}, so G/H is the set of unordered pairs in [n], or the set of edges in the complete graph K_n. m is arbitrary. Now let Γ be a graph with m edges. Any injection from v(Γ) to [n] gives you a point in (G/H)^m, and all such are carried to each other by the action of S_n. Call that orbit [Γ], and let R be Orb – [Γ]. Now a subset S of (G/H)^m is an m-edge graph on n vertices, and S is an R-set just when no m distinct edges form a copy of H. How large can such an S be? That is the Turán number ex(n,Γ), and the Turán problem is to say whatever we can about it.
A lot of problems can be written in this form. (Especially a lot of problems Erdős worked on.) The Heilbronn triangle problem. Problems about subsets of space with forbidden angles. Turán problems for hypergraphs. The Guan-Ramos conjecture and the related Erdős matching conjecture. Bounds for error correcting codes (here take G to be the semidirect product of F_2^n by S_n, and H to be S_n, and m=2, so that Orb is just the set of Hamming distances and the choice of R exactly allows you to exclude whatever distances you want on differences between codewords.) Families of r-dimensional vector spaces of k^n such that any two intersect transversely. The happy ending problem. (G = AGL_2, H = GL_2, m arbitrary, Orb_m = m-tuples of points in the plane up to affine linear equivalence, R = m-tuples not forming a convex polygon.)
Variants
There are many variants of the master question, which allow you to incorporate an even wider range of popular questions under its generous sheltering wings. For instance: you could impose a bound on the size of f(S^m) instead of asking f(S^m) to lie inside a fixed R. (Now you’ve got the Erdős distinct distance problem.) Or instead of a hard constraint you could ask which pairs (|S|, f^{-1}(R)) are possible; the original question asks for which |S| the two entries can be equal. (Now you’ve got the Erdős unit distance problem.) And what do we mean by |S|, anyway? When G is finite, so’s S, but when G is a Lie group (and yes, friends, I do mean either real or p-adic), one had better ask which conditions on R guarantee that S is finite. (Easy exercise: show that a subset of R^2 such that no three points form an angle of less than 0.0001 degrees is finite.) (Harder: the Erdos-Szekeres theorem that R-sets are finite when R is the set of non-convex m-gons that appears in the happy ending problem.) At any rate, there’s no need to insist that S be finite. Maybe “how big can S” be means “how big can its Hausdorff dimension be!” (Still pretty easy exercise: show that a subset of Q_p^* in which no three points form an equilateral triangle has dimension at most log 2 / log p.)
Or, in any one of these cases, you could ask about the chromatic number version of this question: how many pieces do I need to partition (G/H)^m into if each piece P_i has f(P_i) disjoint from R? For sphere packing (O(n) version), this asks: how many pieces of diameter 1 suffice to cover a large sphere? The (spherical) sphere packing number provides an obvious lower bound. This has got to be a known problem, right?
Families
This part is going to be a little more vague and not completely thought out, but I want to write it down so I remember it.
Let me comment that these problems typically come in families. I’m going to be kind of vague about what one should mean by that, because I’m not wholly sure. Take the Turan problem, for example. That’s really a list of problems, one for each n, or at least each sufficiently large n, given by data G_n, H_n, and R_n. But in a way they’re all the same, right? In particular, Orb_m doesn’t change with n (or at least it doesn’t for n large enough) As it happens I am very familiar with this vague notion of seuqences of things, each with an action of S_n, which are somehow all the same — they are functors from the category FI of finite sets with injections, which I’ve written a lot about over the years. In particular, (G_n/H_n) is a finitely generated FI-set, which you can read more about in the paper of Speyer, Ramos, and White. Never mind the definitions; just accept that there’s a reasonable notion of what counts as a family of (G,H,R) instances in this context.
Similarly, I think the cap set problems as n varies should be thought of as a family.
What about the continuous cases? I’m a little less clear there but let me make a stab at at least one kind of thing that should count. Let G be the affine generalized orthogonal group on R^2, i.e. the group of rigid motions where you’re also allowed to dilate, and let H be GO(2). Then G/H is R^2 and Orb_3 is the set of similarity classes of triangles. Write K_n in Orb_3 for the set of triangles such that the ratio between two edges never exceeds n. Then a K_n-set has size bounded by const*n^2 (because a K_n set is more or less the same thing as a packing of unit circles in a circle of radius n.) If U is some OTHER set subset of orb, I would consider a family of problems to be given by R = U intersect K_n, with n growing.
In the first two cases, write (G/H)_n for (G_n/H_n) and in the third case write (G/H)_n for K_n. Note that in all three cases we have an easy asymptotic formula for the maximal size of a ((G/H)_n)-set. Call this size S_n. Then here’s what I’d like to guess.
Vaguely stated guess: In any family of R, if we write m_R(n) for the maximal size of an R-set in the problem indexed by n, then the limit
γ_R = lim_n log(m_R(n) / S_n)
exists. This should be a basic invariant of R that measures how “restrictive” it is.
One could even be more aggressive and ask whether there are constants c_R, γ_R such that
m_R(n) = c_R |S_n|^γ_R + o(|S_n|^γ_R).
which is true in lots of cases where the behavior of m_R(n) is understood!
Is that too abstract? Let me give some sense of what it means. For cap sets, it says the size of the largest cap set in F_q^n grows like q^{γn} or maybe q^{(γ+ε)n} or something. This is known and pretty easy, but what’s not at all known or easy is what γ is. I proved a theorem with Giswijt that γ < 1. I have no idea whether the more aggressive statement is true! I don’t know how to rule out that it’s some kind of funny q^{γ n + epsilon}. For the Turán problem, the guess says log ex(n,Γ) / log n approaches a limit in n; is that known?
If I asked how many points there could be in a unit line segment, no two separated by less than 1/n, and no “approximate three-term APs”: three points x,y,z satisfying y in [(1/2+δ)x + (1/2-δ)z, (1/2-δ)x + (1/2+δ)z], is that maximum asymptotic to some c n^γ, with γ depending on δ? In this context, one might imagine that γ is the largest dimension a set with no δ-approximate 3-term APs could have (throwing out the minimum distance requirement that forces the set to be finite.)
And the aggressive guess would say that, I dunno, if I asked how large a collection of k-element subsets of [n] could be if no m of them shared at least t points, the answer would also be asymptotic to some c n^γ. Believable! I have no reason to believe any of this except that the cases where these problems are solved, e.g. the problems discussed in this survey of set intersection problems, all seem to have answers of this form.
My instinct is that γ_R should be much easier to compute, and that c_R, which is more like a sphere-packing constant, should be more subtle.
Here’s a very concrete question. Take G = S_n, H = S_{n-k}, m = 2. So Orb_2 is just the set of double cosets H \ G / H, which indeed stabilizes for large n to some finite set you can describe combinatorially. For each subset R of this finite set, the guess says there’s some corresponding γ_R, and these numbers are monotone increasing under set inclusion. So… what are they?
I have not thought about this very hard! I am posting with the idea that people will tell me which parts of this are already understood, and which parts are wrong, and in what directions there’s more to say or ask.
Petrichor
Nov. 24th, 2025 11:26 pmEpisode 3:
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