Catherine of Poděbrady and her child live

Catherine was daughter of Czech King George of Poděbrady and wife of Hungarian King Matthias Corvinus, who died in childbirth in 1464. Lets assume she survived and her son too.

What is going to happen then?

Matthias and George were allied at the time, but eventually Matthias turned against his former father-in-law. George went into conflict with the Pope and started to look for Polish support (Casimir IV of Poland also had problems with Papacy because of his conflict with Teutonic Order) and promised to make Casimir's son his successor. But relations between George and Matthias should not deteriorate that much if Catherine lives. George would want to see him as his successor in such scenario and thus Matthias would not turn against him.

Hungary and Bohemia were also claimed by Habsburgs and Jagiellons, so if Matthias holds bot thrones, it should improve relations between HRE Frederick III and Casimir IV of Poland, as Matthias is their common enemy (he was also IOTL, but things were complicated by the fact, that Vladislaus Jagiellon was elected as Czech King, but Habsburgs refused to regognise his title and even ignored him during Imperial election). But while Matthias would have the Emperor against him, he'd still have good relations with the Pope, because Papacy viewed Hungary as shield against Ottoman expansion into Christian lands. And considering the fact, that even IOTL Matthias was able to humiliate Habsburgs, taking Vienna from them, Frederick III is not going to have easy life (especially with Matthias less concerned about succession, as he'd have legitimate adult son).
 
Matthias as King of Bohemia is extremely unlikely in my opinion as his father-in-law had three surviving sons (technically four but the eldest was mentally disabled and so excluded from the succession) so he would have no reason for making a deal about the succession with Catherine‘s husband
 
Matthias as King of Bohemia is extremely unlikely in my opinion as his father-in-law had three surviving sons (technically four but the eldest was mentally disabled and so excluded from the succession) so he would have no reason for making a deal about the succession with Catherine‘s husband
George knew very well that his sons have slim chances to succeede him. His family had too small support base. That is why he selected Vladislaus, who had Polish backing. Matthias would have Hungarian backing, thus he'd be stronger candidate than George's sons. For George it is only realistic chance to keep his bloodline on the throne.
 
Less money wasted on culture. With Beatrice the second wife of Matthias came a lot of Renaissance artists, providing a glamorous court but soaking up royal patronage that would have been better spent on upgrading or building castles against the Ottomans.
 
Matthias and Frederick III already signed a peace treaty and alliance in 1463, although its terms weren't the very best for Matthias. With that being said, Papal support was a cornerstone of Hungarian foreign policy around this time, so I'm not so sure about Matthias siding with his father-in-law over his "adoptive father". He would probably act as a mediator, trying to make the most of the situation in the process. Mathias and Catherine's son becoming the designated heir of Bohemia could be a result of such mediation, while the return of Hungarian borderlands still held by Frederick III could be another.

One thing is sure, Matthias ITTL wouldn't get entangled in Western adventures like he was IOTL. I would expect there being more frequent and more intense wars against the Ottomans instead. Matthias might even be able to enlist monetary or military support from both Austria and Bohemia. Meanwhile, the active anti-Turkish orientation would please not only the Papacy, but also Matthias' domestic powerbase (like the Archbishop of Esztergom, John Vitéz).
Less money wasted on culture. With Beatrice the second wife of Matthias came a lot of Renaissance artists, providing a glamorous court but soaking up royal patronage that would have been better spent on upgrading or building castles against the Ottomans.
Matthias was a true "Renessaince Prince" long before Beatrix even stepped foot in the country. He was well-read in the Classics, an admirer of Humanism and patron of arts. The "import" of Italian Humanists and the "export" of Hungarian students to Western universities was already an established fact of the 1460s. It's true however, that Beatrix's absence would keep the Court less extravagant and ceremonial, therefor less expensive. Beatrix also introduced something of a wedge between Matthias and his subjects, so that would be gone too ITTL.
 
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