Red Bull took four straight double titles at the start of the decade but has taken only a handful of wins in most years of the hybrid era, frequently pinning its deficit on erstwhile supplier Renault.
Verstappen, the former teammate of Australian Daniel Ricciardo, is set to be among Hamilton's chief challengers this year, and the young Dutchman put his new auto through its paces on Wednesday.
The 2019 Red Bull auto is the first to be made since the former world champions formed a partnership with new power providers Honda.
Red Bull has revealed the first images of its RB15 for the 2019 F1 season in a dramatic "one-off livery".
"Progress is obviously our target", Horner said.
Max Verstappen gave the auto its first laps in a shakedown at Silverstone, the first time the team have run a Honda engine. We've obviously been able to strike at venues that have suited the characteristics of our auto, winning four races last year - should have been five - and three races the year before.
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"I am very excited to get started as the off-season has been quite long", said Verstappen, 21. "And from there onwards, you just start to work together with the vehicle [to see] what you can improve".
Australian Daniel Ricciardo has left Red Bull to join Renault with Verstappen, who finished fourth in the drivers' standings previous year, to be partnered by young Frenchman Pierre Gasly. "It was a very positive day so very happy about it".
"At the moment I am of course very optimistic, but also realistic". But then after, for sure, we need to give them time as well.
"Formula 1 is the pinnacle of motorsport". You fight against teams like Ferrari, Mercedes, Renault and when you start a new partnership, it always takes time before things run super smoothly and in the most efficient way.
What do you make of the new Red Bull challenger and do you think Honda will help them scale new heights this season?