If the wearer is a cavalier, these vambraces display the symbol of his order. When unworn or in the hands of a non-cavalier, they appear to be an ordinary pair of steel vambraces.
A cavalier wearing these vambraces is considered two class levels higher for purposes of determining the effects of his challenge class feature. Furthermore, once per day as a swift action, a cavalier with the tactician class feature can increase the area in which he grants teamwork feats. Before the end of his next turn, when the cavalier grants his allies a teamwork feat, he can grant it to all allies within 60 feet, though they must still be able to see and hear him.
Challenge has three level dependent properties1:"The cavalier’s melee attacks deal extra damage whenever the attacks are made against the target of his challenge. This extra damage is equal to the cavalier’s level."
2:The cavalier can use this ability once per day at 1st level, plus one additional time per day for every three levels beyond 1st, to a maximum of seven times per day at 19th level."
3:The bonus (or penalty in a handful of cases) to challenge gained from one's order, which typically "increases by 1 for every four levels the cavalier possesses."
I can see three ways to argue this
A: They're all increased
B: Only 1 and 2 increase, as they're part of the challenge feature while 3 is part of order.
C: 1 or 1+3 increase because they are effects. 2 does not because it's not an effect of the challenge but a usage limit.
Champion's Banner is similar while explicitly saying 2 isn't increased. Is this reminder text or exception? The banner is slotless for the same price, which should mean it's less powerful, but strictly better items with unusual effects aren't unheard of. It also still isn't clear on if 3 increases.
A champion’s banner is a cloth flag or standard (typically 2 feet wide and 4 feet long) meant to be carried and displayed on a lance, polearm, frame, or staff. It has no effect unless it is mounted properly and a wielder is carrying the object bearing it. It depicts the insignia or heraldic symbol of its wielder; if the wielder has no such symbol, it depicts a heraldic lion.
A character with the challenge class feature carrying a champion’s banner treats his class level as 4 levels higher when determining the effect of his challenge ability. This does not alter the number of times per day the wielder may use his challenge ability, only the value of the bonuses it grants when he does so.
RAW a class feature is a class feature, not the other things that touch that class feature.
The challenge class feature says:
Challenge (Ex): Once per day, a cavalier can challenge a foe to combat. As a swift action, the cavalier chooses one target within sight to challenge. The cavalier’s melee attacks deal extra damage whenever the attacks are made against the target of his challenge. This extra damage is equal to the cavalier’s level. The cavalier can use this ability once per day at 1st level, plus one additional time per day for every three levels beyond 1st, to a maximum of seven times per day at 19th level.
Challenging a foe requires much of the cavalier’s concentration. The cavalier takes a –2 penalty to his Armor Class, except against attacks made by the target of his challenge.
The challenge remains in effect until the target is dead or unconscious or until the combat ends. Each cavalier’s challenge also includes another effect which is listed in the section describing the cavalier’s order.
The items under the challenge class feature are.- extra damage dependant on the cavalier level;
- number of attacks in a day;
- "the other effect listed in the section describing the cavalier’s order", as that is part of the feature.