UBI - Raising the Poverty line Imagine if a country starts to pay $1000/month to all citizens.
Then my landlord knows that I can afford to pay a bit more rent from next month.
So will my favorite ice-cream place, my barber and in general the entire market around me raises it's price because of the theory of tragedy of commons and everyone would want a bigger piece of pie from my income.
Some of the proponents of UBI argue that we would effectively remove poverty with UBI, but with the logic above, it seems like a snowball effect will make everything more expensive and nullify the effect  of UBI to remove poverty by causing inflation.

When everything becomes more expensive, wouldn't it just raise the poverty line keeping the poor still in need of support?
Is there any research done on this subject?

P.S: I realize this is home-cooked economics, I have never studied economics academically so excuse me if the solution is fairly obvious but I can't see it and Googling it isn't helping as I don't think I know the right terms that I should lookup.
No, this question is based on fundamentally flawed premises.  There is a plenty that you can criticize about UBI and generally literature does not consider UBI efficient redistribution policy, but none of what is mentioned in the question holds for UBI.
Addressing UBI

Lets start by correcting misconceptions:

Theory of tragedy of commons has nothing to do with what you describe. The theory of tragedy of commons is:
a) A theory about situation where private property rights are non-existent. Unless you want to make claim that property rights are missing in the market for ice cream, barber services or entire market generally suffers from lack of property rights - which is empirically absurd assertion outside few specific markets or countries like North Korea - tragedy of the commons simply does not apply here as for tragedy of the commons you first have to have some commons to begin with.
b) the tragedy of the commons is not about increases in prices. In fact, under tragedy of the commons prices will tend to be lower, or in some cases even impossible to exist without further government intervention (e.g. clean air is an example of resource suffering from tragedy of commons) than they would be in presence of property rights. Furthermore, the tragedy of the commons will generally cause prices to be undesirably low (from societal perspective) which will result in overexploitation of common resource.

UBI Should not Cause Significant Inflation:

First, it is important to understand that profit maximizing firms can't just set prices in arbitrary fashion. They are limited by the parameters of the market such as the level of competition or market structure we are talking about. Inflation, is generally unaffected by redistribution.
Inflation is macroeconomic phenomenon and depends on the equilibrium price level set on the money market. The money market equilibrium can be in it's simples form described by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where M is the money supply, V velocity of money, P price level and Y output.
Solving for price level and log-linearizing we get:
$$\ln P=\ln M+\ln V−\ln Y$$
Thanks to everything being in logs we can directly interpret any change in $P$ (i.e. inflation if change is positive or deflation if negative) in response to right hand side variables as percentage changes. So if money supply would expand by $1\%$ we would expect inflation to increase by $1\%$ ceteris paribus.
However, UBI does not directly affect the right hand side variables.
a) Provided that UBI is funded by taxes not high powered money it does not affect money stock of economy at all. So it does not affect $M$ at all.
b) Velocity of money is quite constant in a long-run, in short-run it depends mainly on the business cycle and interest rates. There is no reason to suspect UBI would have any effect.
c) UBI is a redistribution measure so it transfers output from one part of economy into another. To the extent this is funded by distortionary taxation it reduces economic efficiency and output in an economy which has no market imperfections. However, real life economies contain externalities, missing markets and other issues that can create efficiency cause for redistribution (see discussion of this in Atkinson (2015) Inequality). Generally, empirical literature shows that too large levels of redistribution can lead to lower growth of output, but modest redistribution seems to have little to none impact on it (see Jonathan, Ostry and Berg: Redistribution, Inequality, and Growth). Consequently, unless the UBI would be set to some extremely large value it should have no significant effect on $Y$ and thus again no effect on inflation.
UBI could distort relative prices because while everyone receives the UBI only some people pay for it. To the extent that the spending patterns of people who receive net transfer and those who on net pay into the system relative prices could change in a way that would hurt poor even if inflation remains overall constant. This would happen because it would lead to increase in demand for goods that low-income people consume while simultaneously decreasing demand for items that high income people consume. However, this effect would happen with any redistributive measure, and in addition there is no reason to believe this effect would be large enough to offset real increase in welfare of the poor thanks to the redistribution.
Literature on UBI
For overview of literature on this you can see Hoynes and Rothstein (2019) or Martinelli (2017) and sources cited therein. As mentioned, at the beginning, literature is generally negative towards UBI but that is because research shows it is less efficient and more wasteful than traditional redistribution systems, not because it would lead to inflation.