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You definitely need the trailing \007 (or \033\\) or some terminals will stop responding waiting for the end of the sequence. Did you try this inside tmux? tmux has a timeout to prevent panes hanging, but most other terminals do not. If you try it in xterm without the \007, it will hang.

For information on the title setting sequences themselves, look at https://invisible-island.net/xterm/ctlseqs/ctlseqs.html (title setting is OSC 0 and 2 - look under "Operating System Commands").

ts and fs are termcap names (terminfo is tsl/fsl) meaning "to status line" and "from status line" (as the name implies, infocmp is showing you terminfo, not termcap). These are somewhat historical from the days when hardware terminals had a separate status line, but in recent years have been reused sometimes for the X terminal title. Look at the "Status Lines" section in terminfo(5) for some bits about status line capabilities. In practice, tsl and fsl are not often provided or used and instead most applications look at the XT capability ("xterm titles") to work out if they should use the OSC sequences. XT is an extension capability - you need to use "infocmp -x" to see it.

You can always set the title inside tmux using the OSC sequences, by default the title (#{pane_title} format) is shown in quotes in status-right. tmux will not also set the external terminal title (the terminal tmux is running in) unless the set-titles option is on. The set-titles-string option controls the format of the title tmux sets (by default it includes the active pane's title and some other information). set-titles will only work if TERM outside tmux specifies an entry that contains XT or tsl/fsl (xterm does have XT).

I don't know how to configure vim to set the title but it seems like you are on the right track there with t_ts and t_fs - it seems like vim is using the tsl/fsl but using the termcap names (probably a legacy from vi which was started long before terminfo existed).

I can give you a brief description of how it works:

• termcap was originally written in the 70s to allow vi to run on different (hardware) terminals;

• terminfo was written as a replacement in the 80s to fix a number of problems and limitations with termcap;

• both are still used (terminfo much more commonly) for both software and hardware terminals;

• they both work in a similar way - there is a database of named entries (like "xterm" or "tmux-256color") which give a set of named capabilities of the terminal; some capabilities are flags (eg "XT") or numbers (eg "colors") but most are strings which specify the sequence an application should send to a terminal to make something happen (eg "cuu1" make the cursor move one line up - "\033[A" on vt100-compatible terminals);

• termcap has short most two-character names, terminfo tend to be longer; termcap has a small limit of 1024 bytes on the length of a database entry so it is often missing many capabilities that terminfo has;

• TERM is set to the name of the termcap or terminfo entry the application should look for in the database;

• the terminfo database is maintained and shipped as part of ncurses (a library for drawing to the terminal which is used by many terminal applications); the termcap database is generated from terminfo using tic.

There isn't really a single source with all the information put together. For the escape sequences, https://vt100.net/ has a lot of manuals for old hardware terminals (much of which is now irrelevent however) and the link I gave above for xterm documents the escape sequences supported by xterm and most other modern terminal emulators support a subset of these (almost all terminals still around are vt100-compatible and xterm is the de facto standard terminal). ncurses has some manuals: terminfo(5) and termcap(5). There is an O'Reilly book called "termcap and terminfo" which covers much as well, but it was written in the late 80s so some of it is out of date.