Prove that every triangle-free graph is  $\lceil 2\sqrt{n}\rceil$ colorable
I'm struggling with the following question;     For every graph $G$ such that $K_3 \not\subseteq G$ (i.e. $G$ does not contain a triangle), prove that $\chi(G) \leq 2\sqrt{n} +1$ (where $\chi(G)$ denotes the chromatic number).   Any hints? I really cannot see the connection between the triangle-free nature of $G$ and a chromatic number being bounded above by $O(\sqrt{n})$.
How do I write a URL in Markdown on Stack Exchange? Which is easier, pushing or pulling? It is generally assumed, from a person's perspective, that pushing a cart is more easier than pulling one. But why? Is there any difference in terms of force required to achieve the same amount of displacement?  Or is it merely human perception?  Why is it that almost all automobiles transfer torque to the back axle. But then, why do trains have engines in the front? In TeX, how is computing zero sign written with slanted vertical line on it? Applying Dupli-Objects Is it possible to bake dupliverts into one mesh for the BGE? (I have a large ground landscape that I use as the parent to duplicate trees, and the BGE doesn't support most modifiers.) Is conservation of momentum and energy valid for non-inertial frames? In Clash of Clans there was an update on 9/16 adding a Tie-breaker into the wars, can anybody explain in detail how this mechanism works? What prevents investors from buying high yield stocks and selling them as soon as their dividend is paid out? What prevents investors from buying high yield stocks and selling them as soon as their dividend is paid out?   I guess I am confused: If you know the yield and when the dividend will be paid out, why not buy a share an instant before the dividend is paid out and then sell it?  Do you have to have held the share for some amount of time to get the dividend? Is there too much volatility in the instant the dividend is paid out? Why does observation collapse the wave function? In one of the first lectures on QM we are always taught about Young's experiment and how particles behave either as waves or as particles depending on whether or not they are being observed. I want to know what about observation causes this change? For creating the graphics for my thesis (function graphs, flow charts, line art sketches) I've chosen to use the sans serif font by  \renewcommand\familydefault{\sfdefault}    However when I use mathematical symbols like $p_{s}$, they are still typeset in roman font and that looks weird:    So I wonder:   how could I force mathematical expressions also be typeset in \sfdefault if I do, are there any negative sideeffects to expect (with greek signs or other special symbols?) OR should I typeset all "mathematical" graphs (axis labels etc.) in roman font, and only make the other graphics like flow charts and sketches with sans serif font? Find all functions $f:\mathbb{N_{0}} \rightarrow \mathbb{N_{0}}$ satisfying the equation $f(f(n))+f(n)=2n+3k,$ for all &amp; $n\in \mathbb{N_{0}}$, where $k$ is a fixed natural number.  A friend of mine suggested me to use recurrence relation. While it is unfortunate that I have problems understanding recurrence relation. Please refer information to understand difference equation, homogeneous equation &amp; auxillary equation.  Thanks in advance! Kindly don't down vote for not showing my attempt. I am seriously not able to understand the question and the process suggested to use. I am using lightning:recordEditForm and lightning:inputField for creating a new record. this component is being used inside a lightning community builder. When i click the submit button, i am able to create a new record in my custom object. However i need the created record ID as i need to pass it in the URL for the next page. Partition around bad drive sectors? I've seen a lot of hard drives with only a few bad (unrecoverable) sectors (as indicated by SMART). Of course the proper thing to do with a bad drive is replace it, as there's a good chance of the drive crashing completely in the near future; I'm aware of that. However, in the interim, or for those who don't want to spend $70 to save data they don't value highly anyway (sigh, non-technical people), or just to get some use out of old hardware instead of throwing it away, I'm wondering if there's a way to identify where the unrecoverable sectors are ("where" meaning numerically, from 0 to the number of sectors on the drive), so I could stick a blank partition in that general area. Yes, this would involve low-level-formatting the drive and reinstalling the OS, of course.  So:  Would partitioning around unrecoverable sectors even be a workable way of reducing the disk of losing data?  If so, is there a way to find the location of these unrecoverable sectors? How to configure an exposed filter in a view using a date picker? Upgrade to the latest PHP version in Ubuntu I am using  in my server and NextCloud informs me that:     You are currently running PHP 7.0.32-0ubuntu0.16.04.1. Upgrade your   PHP version to take advantage of performance and security updates   provided by the PHP Group as soon as your distribution supports it.   So, I tried to upgrade me OS (Ubuntu 16.04), by executing:  apt-get update  apt-get dist-update  Then I restarted the server and I re-executed the above commands. After all these, when I connect from terminal I am getting:  Welcome to Ubuntu 16.04.5 LTS (GNU/Linux 4.4.0-83-generic x86_64)   * Documentation:  https://help.ubuntu.com  * Management:     https://landscape.canonical.com  * Support:        https://ubuntu.com/advantage  0 packages can be updated. 0 updates are security updates.   and my PHP version is still PHP 7.0.32-0ubuntu0.16.04.1 (cli) ( NTS ).  This means that PHP 7.0.32 is the latest possible version that I can install in ubuntu 16.04; Using Integration By Parts results in 0 = 1 I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation.  I start with the following equation:  $$\int \frac{1}{f} \frac{df}{dx} dx$$  I let:  $$u = \frac{1}{f} \text{ and } dv = \frac{df}{dx} dx$$  Then I find:  $$du = -\frac{1}{f^2} \frac{df}{dx} dx \text{ and } v = f$$  I can then substitute into the usual IBP formula:  $$\int udv = uv - \int v du$$  $$\int \frac{1}{f} \frac{df}{dx} dx = \frac{1}{f} f - \int f \left(-\frac{1}{f^2} \frac{df}{dx}\right) dx$$  $$\int \frac{1}{f} \frac{df}{dx} dx = 1 + \int \frac{1}{f} \frac{df}{dx} dx$$  Then subtracting the integral from both sides, I've now shown that:  $$0 = 1$$  Obviously there must be a problem in my derivation here...  What wrong assumption have I made, or what error have I made?  I'm baffled. The easiest thing to do is [to] call for a taxi Show that $f$ is differentiable on $(−1, 1)$. Let $f$ be a function defined on the interval $(−1, 1)$ such that for all $x, y \in (−1, 1)$, $$f(x + y) = \frac{{f(x) + f(y)}}{{1 - f(x)f(y)}}.$$ Suppose that $f$ is differentiable at $x = 0$, show that $f$ is differentiable on $(−1, 1)$.  Need your kind guidance and help. Thanks in advance! I am looking for an option to require a passcode to power off my iPhone. That way, if the phone is lost, the person who finds it cannot simply turn it off, rendering Find My iPhone useless. Is there a way to accomplish this? White space below footer I would like to show the author names using \citet but I get just the reference number. Here is a minimal code:  \documentclass{report} \usepackage[utf8]{inputenc} \usepackage[english]{babel} \usepackage{natbib}  \begin{document}   The author \citet{toschi09} has shown that ..  \bibliographystyle{plain} \bibliography{foo}  \end{document}   And here is foo.bib  @article{toschi09, author = {Toschi, F. and Bodenschatz, E.}, title = {Lagrangian Properties of Particles in Turbulence}, journal = {Annual Review of Fluid Mechanics}, volume = {41}, number = {1}, pages = {375-404}, year = {2009},  }   What I get is:     The author (author?) [1] has shown that ..   Any help will be more than welcome, thank you!